- Example 1 - Solve the equation \(\log_\frac{1}{5} x = -2\)
Solution - \begin{eqnarray}\log_\frac{1}{5} x &=& -2\\\nonumber x &=& \left(\frac{1}{5}\right)^{-2}\\\nonumber x &=& 25\end{eqnarray} - Example 2 - Solve the equation \(\log_x \frac{1}{16} = 2\)
Solution - \begin{eqnarray}\log_x \frac{1}{16} &=& 2\\\nonumber x^2 &=& \frac{1}{16}/^\sqrt{}\\\nonumber x_{1,2} &=& \pm \left(\frac{1}{4}\right)\end{eqnarray} - Example 3 - Solve the equation \(10^x = 111\)
Solution - \begin{eqnarray}10^x &=& 111/\log\\\nonumber x &=& \log 111 = 2.04532\end{eqnarray} - Example 4 - Solve the equation \(\log(5x) + \log(2x+3) = 1 + 2 \log(3-x)\)
Solution - \begin{eqnarray}\log(5x) + \log(2x+3) &=& 1 + 2 \log(3-x)\\\nonumber \log(5x(2x+3)) &=& \log 10 + \log(3-x)^2 \\\nonumber \log(5x(2x+3)) &=& \log(10(3-x)^2)\\\nonumber 10x^2 +15x &=& 10(9-6x+x^2)\\\nonumber 10x^2 + 15x &=& 90 - 60x + 10x^2\\\nonumber 75x &=& 90/:75\\\nonumber x &=& \frac{90}{75} = \frac{6}{5}\end{eqnarray} - Example 5 - Solve the equation \(\frac{\log_3 x-1}{7+\log_3 x} + \frac{\log_3 x + 7}{1-\log_3 x} = 0\)
Solution - To solve this equation the substitution in form of \(\log_3 x = t\) is introduced. \begin{eqnarray}\frac{\log_3 x-1}{7+\log_3 x} + \frac{\log_3 x + 7}{1-\log_3 x} &=& 0\Rightarrow \log_3 x = t \\\nonumber \frac{t-1}{7+t}+\frac{t+7}{1-t} &=& 0 /\cdot(7+t)(1-t) \\\nonumber -(t-1)(t-1) + (t+7)^2 &=& 0 \\\nonumber -(t-1)^2 &=& -(t+7)^2/\cdot(-1) \\\nonumber t^2 -2t + 1 &=& t^2 +14t + 49 \\\nonumber -16t &=& 48/:(-16)\\\nonumber t &=& -3\end{eqnarray} Now that the value of variable t is determined the value of variable x can be determined by substituting value of \(t\) into the equation \(\log_3 x = t\). \begin{eqnarray}\log_3 x &=& -3 \\\nonumber x &=& 3^{-3} \\\nonumber x &=& \frac{1}{27}\end{eqnarray} - Example 6 - Solve the equation \(\log_3(5+4\log_2(x-1)) = 2\)
Solution - \begin{eqnarray}\log_3(5+4\log_2(x-1)) &=& 2\\\nonumber 5 + 4 \log_2(x-1) &=& 9\\\nonumber 4\log_2(x-1)&=& 4/:4\\\nonumber \log_2(x-1)&=& 1\\\nonumber x-1 &=& 2\\\nonumber x &=& 3 \end{eqnarray} - Example 7 - Solve the equation \(\log_3(2^x - 7) = 2\)
Solution - \begin{eqnarray}\log_3(2^x - 7) &=& 2\\\nonumber 2^x - 7 &=& 9 \\\nonumber 2^x &=& 16\\\nonumber x &=& 4\end{eqnarray} - Example 8 - Solve the equation \(x^{1+2\log x} = 100x^{\log x}\)
Solution - \begin{eqnarray}x^{1+2\log x} &=& 100x^{\log x}/\log\\\nonumber (1+2\log x)\log x &=& \log (100x^{\log x}) \\\nonumber \log x + 2\log^2x &=& \log 100 + \log x^{\log x}\\\nonumber \log x + 2\log^2x &=& 2 + \log^2 x\\\nonumber \log^2x + \log x - 2 &=& 0\end{eqnarray} To solve the previous equation the substitution is introduced in which \(\log x \) is substituted with variable \(t\) (\(\log x = t.\)) \begin{eqnarray}t^2 + t - 2 &=& 0 \\\nonumber t_{1,2} = \frac{-1 \pm \sqrt{1+8}}{2} & = & \frac{-1\pm3}{2}\\\nonumber t_1 = \frac{-1 + 3}{2} &=& 1 \\\nonumber t_2 = \frac{-1-3}{2} &=& -2\end{eqnarray} The values of variable \(t\) are determine the next step is to determine the values of variable \(x\) using the substitution equation \(\log x = t\)\begin{eqnarray}\log x_1 &=& t_1 \Rightarrow \log x_1 = 1 \Rightarrow x_1 = 10 \\\nonumber \log x_2 &=& t_2 \Rightarrow \log x_2 = -2 \Rightarrow x_2 = 10^{-2}\end{eqnarray} - Example 9 - Solve the equation \(e^{2x} = e^x + 6\)
Solution - \begin{eqnarray}e^{2x} &=& e^x + 6\\\nonumber e^{2x} - e^x -6 &=&0\end{eqnarray} This equation can be transformed into quadratic equation by introducing the substitution \(e^x = t\). The equation can be written in the following form $$ t^2 - t - 6 =0$$ The solution of quadratic equation can be written as \begin{eqnarray} t_{1,2} &=& \frac{1 \pm \sqrt{1+4\cdot 6}}{2} \\\nonumber t_{1,2} &=& \frac{1\pm 5}{2}\\\nonumber t_1 &=& \frac{1+5}{2} = \frac{6}{2} = 3 \\\nonumber t_2 &=& \frac{1-5}{2} = -2\end{eqnarray} The value of variable t was determined by solving the quadratic equation. The values of variable \(x\) are determine by substituiting the value of \(t\) into the equation \(e^x = t.\) \begin{eqnarray} t_1 &=& 3 \Rightarrow e^{x_1} = t_1 \Rightarrow e^{x_1} = 3/\ln\\\nonumber \ln e^{x_1} &=& \ln 3 \Rightarrow x_1 = \ln 3 \end{eqnarray} The other solution \(t_2 = -2\) is not real since the output value of exponential value can not be negative value. - Example 10 - Solve the equation \(10^{3x-1} = 2^{2x+1}\)
Solution - \begin{eqnarray}10^{3x-1} &=& 2^{2x+1}/\log \\\nonumber (3x-1) \log 10 &=& (2x+1) \log 2 \\\nonumber 3x-1 &=& 2x\log2 + \log 2 \\\nonumber 3x-2x\log 2 &=& 1+ \log 2\\\nonumber x(3-2\log 2) &=& \log 2 + 1 /: (3-2\log2) \\\nonumber x &=& \frac{\log 2 +1 }{3-\log2}\end{eqnarray} - Example 11 - Solve the equation \(\log_x 2 - \log_4 x + \frac{7}{6} = 0\)
Solution - \begin{eqnarray}\log_x 2 - \log_4 x + \frac{7}{6} &=& 0\end{eqnarray} Text missing \begin{eqnarray} \frac{\frac{1}{2}\log_2 2}{\log_4 x} - \log_4 x + \frac{7}{6} &=& 0\\\nonumber \frac{1}{2\log_4 x} - \log_4 x +\frac{7}{6} &=& 0 \Rightarrow \log_4 x = t\end{eqnarray} \begin{eqnarray}\frac{1}{2t} - t + \frac{7}{6} &=& /\cdot 6t \\\nonumber 3 - 6t^2 + 7t &=& 0 /\cdot(-1) \\\nonumber 6t^2 - 7t -3 &=& 0 \end{eqnarray} \begin{eqnarray}t_{1,2} &=& \frac{7\pm\sqrt{49 + 4\cdot 3 \cdot 6}}{12}\\\nonumber t_{1,2} &=& \frac{7 \pm 11}{12}\\\nonumber t_1 = \frac{7+11}{12} = \frac{18}{12} = \frac{3}{2} \\\nonumber t_2 &=& \frac{7 - 11}{12} = -\frac{4}{12} = -\frac{1}{3}\end{eqnarray} The values of variable \(t\) are determine and the next step is to determine the values of variable \(x\) by substituting values of \(t\) into the equation \(\log_4 x = t\) \begin{eqnarray}t_1 &=& \frac{3}{2} \Rightarrow \log_4 x_1 = \frac{3}{2}\\\nonumber x_1 &=& 4^\frac{3}{2} \Rightarrow x_1 &=& \sqrt{\left(2^2\right)^3} = 2^3 = 8\\\nonumber t_2 &=& -\frac{1}{3} \Rightarrow \log_4 x_2 = -\frac{1}{3} \\\nonumber x_2 &=& 4^{-\frac{1}{3}}\end{eqnarray} - Example 12 - Solve the equation \(\begin{cases}x+y = 34 \\ \log_2 x + \log_2 y = 6\end{cases}\)
Solution - \begin{eqnarray}\begin{cases}x+y = 34\Rightarrow x = 34-y \\ \log_2 x + \log_2 y = 6\end{cases}\\\nonumber \log_2 (xy) &=& 6\\\nonumber \log_2 (34-y)y &=& 6 \\\nonumber 34y-y^2 - 64 &=& 0/\cdot(-1) \\\nonumber y^2 - 34y + 64 &=& 0\end{eqnarray} Text missing \begin{eqnarray} y_{1,2} &=& \frac{34 \pm \sqrt{34^2 - 4 \cdot 64}}{2} = \frac{34\pm 30}{2}\\\nonumber y_1 &=& \frac{34 + 30}{2} = \frac{64}{2} = 32 \\\nonumber y_2 &=& \frac{34-30}{2} = 2\end{eqnarray}Text missing\begin{eqnarray} y_1 &=& 32\Rightarrow x_1 = 34 - 32 = 2 \\\nonumber y_2 &=& 2 \Rightarrow x_2 = 34 - 2 = 32\end{eqnarray} - Example 13 - Solve the equation \(7\cdot 2^x - 4^x = 12 \)and multiply the equation solutions.
Solution - \begin{eqnarray}7\cdot 2^x - 4^x &=& 12\\\nonumber 7\cdot 2^x - 2^{2x} &=& 12 \Rightarrow 2^x = t \\\nonumber 7t - t^2 -12 &=& 0 /\cdot(-1) \\\nonumber t^2 -7t + 12 &=& 0\\\nonumber t_{1,2} &=& \frac{7\pm \sqrt{49 - 2\cdot 12}}{2}\\\nonumber t_{1,2} &=& \frac{7 \pm 1}{2} \\\nonumber t_1 &=& 4 \\\nonumber t_2 &=& 3\\\nonumber 2^{x_1} &=& 2^2 \\\nonumber x_1 &=& 2 \\\nonumber 2^{x_2} = 3 \log_2 \\\nonumber x_2 &=& \log_2 3\end{eqnarray} \begin{eqnarray}x_1\cdot x_2 &=& 2\log_2 3 = \log_2 9\end{eqnarray} - Example 14 - Solve the equation \(\log_7 x = 3\)
Solution - \begin{eqnarray}\log_7 x &=& 3 \\\nonumber x &=& 7^3 = 7 \cdot 7 \cdot 7 = 343\end{eqnarray} - Example 15 - Solve the equation \(\log_2(x+3) + \log_2 (x+2 ) = 1\)
Solution - \begin{eqnarray}\log_2(x+3) + \log_2 (x+2 ) &=& 1 \\\nonumber \log_2(x^2 + 2x+ 3x + 6) &=& \log_2 2\\\nonumber x^2 + 5x + 6 &=& 2 \\\nonumber x^2 + 5x + 4 &=& 0 \\\nonumber x_{1,2} &=& \frac{-5 \pm \sqrt{25-4\cdot 4}}{2} = \frac{-5 \pm 3}{2}\\\nonumber x_1 &=& \frac{-5+3}{2} = -1 \\\nonumber x_2 &=& \frac{-5-3}{2} = -4\end{eqnarray} - Example 16 - Solve the equation \(9^x - 5\cdot 3^x + 4 = 0\)
Solution - \begin{eqnarray}9^x - 5\cdot 3^x + 4 &=& 0\\\nonumber 3^{2x} - 5 \cdot 3^x + 4 &=& 0 \Rightarrow 3^x = t \\\nonumber t^2 - 5t + 4 &=& 0 \\\nonumber t_{1,2} &=& \frac{5 \pm \sqrt{25 - 4\cdot 4}}{2} = \frac{5 \pm 3}{2} \\\nonumber t_1 &=& \frac{5+3}{2} = 4 \\\nonumber t_2 &=& \frac{5-3}{2} = 1\\\nonumber t_1 = 4 \Rightarrow 3^{x_1} &=& t_1 \Rightarrow 3^{x_1} = 4 /\log_3 \Rightarrow x_1 = \log_3 4\\\nonumber t_2 = 1 \Rightarrow 3^{x_2} &=& t_2 \Rightarrow 3^{x_2} = 1/\log_3 \Rightarrow x_2 = 0\end{eqnarray} - Example 17 - Solve the equation \(\log_7 x = 2 \log_7 2 + 3\log_7 3 \)
Solution - \begin{eqnarray}\log_7 x &=& 2 \log_7 2 + 3\log_7 3\\\nonumber \log_7 x &=& \log_7 2^2 + \log_7 3^3\\\nonumber \log_7 x &=& \log_7 4 + \log_7 27\\\nonumber \log_7 x &=& \log_7 108 \\\nonumber x &=& 108 \end{eqnarray} - Example 18 - Solve the equation \(\log_2(3-2^x) = 1-x\)
Solution - \begin{eqnarray}\log_2(3-2^x) &=& 1-x \\\nonumber 3 -2^x &=& 2 \cdot 2^{-x}/2^x \\\nonumber 3\cdot 2^x - 2^{2x} = 2/\cdot(-1)\\\nonumber 2^{2x} - 3\cdot 2^x + 2 &=& 0 \Rightarrow 2^x = t \\\nonumber t^2 - 3t +2 &=& 0 \\\nonumber t_{1,2} &=& \frac{3 \pm \sqrt{9-4\cdot 2}}{2}\\\nonumber t_{1,2} &=& \frac{3 \pm 1}{2} \\\nonumber t_1 &=& \frac{3+1}{2} = \frac{4}{2} = 2\\\nonumber t_2 &=& \frac{3-1}{2} = \frac{2}{2} = 1 \end{eqnarray} After the unkonw t values are determined the next step is to determine the unknown values of \(x\) using the substitution expression \(2^x = t.\)\begin{eqnarray} 2^{x_1} &=& 2/\log_2 \Rightarrow x_1 = 1\\\nonumber 2^{x_2} &=& 1/\log_2 x_2 = 0\end{eqnarray} Sum of the obtained solutions \(x_1+x_2 = 1 + 0 = 1\). The product of obtained solutions \(x_1\cdot x_2 = 0\) - Example 19 - Solve the equation \(\log_2 (x+1)- \log_2(2x-1) = 1\)
Solution - \begin{eqnarray}\log_2 (x+1)- \log_2(2x-1) &=& 1\\\nonumber \log_2\left(\frac{x+1}{2x-1}\right) &=& \log_2 2\\\nonumber x+1 &=& 4x -2 \\\nonumber -3x &=& -3 /:(-3) \\\nonumber x &=& 1\end{eqnarray} - Example 20 - Solve the equation \(2\log(3x-5) - \log(x+1) = 2-2\log 5\)
Solution - \begin{eqnarray}2\log(3x-5) - \log(x+1) &=& 2-2\log 5 \\\nonumber \log(3x-5)^2 - \log(x+1) &=& \log 10^2 - \log 5^2 \\\nonumber \log \left(\frac{9x^2 - 30 x + 25}{x+1}\right) &=& \log\left(\frac{100}{25}\right)\\\nonumber \frac{9x^2 -30x +25}{x+1} &=& 4 /\cdot(x+1) \\\nonumber 9x^2 - 30x +25 &=& 4x + 4 \\\nonumber 9x^2 -34x +25 &=& 0 \\\nonumber x_{1,2} &=& \frac{34 \pm \sqrt{34^2 - 4\cdot 21\cdot 9}}{18}\\\nonumber x_{1,2} &=& \frac{34 \pm 20}{18}\\\nonumber x_1 &=& \frac{34 + 20}{18} = \frac{54}{18} = 3 \\\nonumber x_2 &=& \frac{34-20}{18} = \frac{14}{18} = \frac{7}{9}\end{eqnarray} - Example 21 - Solve the equation \(\log_2 (\log_4 x) = 1\)
Solution - \begin{eqnarray}\log_2 (\log_4 x) &=& 1\\\nonumber \log_4 x &=& 2 \\\nonumber x &=& 4^2 \\\nonumber x&=& 16\end{eqnarray} - Example 22 - Solve the equation \(\log_2(3-2^{-x} = 1-x\)
Solution - \begin{eqnarray}\log_2(3-2^{-x} &=& 1-x\\\nonumber 3-2^{-x} &=& 2^{1-x}/\cdot 2^x\\\nonumber 3\cdot 2^x - 1 &=& 2 \\\nonumber 3 \cdot 2^x &=& 3 /:3 \\\nonumber 2^x &=& 2^0 \Rightarrow x = 0 \end{eqnarray} - Example 23 - Solve the equation \(\log^2 x - \log x^2 - 8 = 0\)
Solution - \begin{eqnarray}\log^2 x - \log x^2 - 8 &=& 0 \\\nonumber \log^2 x - 2\log x - 8 &=& 0 \Rightarrow \log x = t \\\nonumber t^2 - 2t - 8 &=& 0 \\\nonumber t_{1,2} &=& \frac{2 \pm \sqrt{4 + 4\cdot 8}}{2}\\\nonumber t_{1,2} &=& \frac{2\pm 6}{2}\\\nonumber t_1 &=& \frac{2+6}{2} = 4\\\nonumber t_2 &=& \frac{2-6}{2} = -2 \\\nonumber \log x_1 = t_1 \Rightarrow \log x_1 &=& 4 \Rightarrow x_1 = 10^4 = 10000\\\nonumber \log x_2 = t^2 \Rightarrow \log x_2 &=& -2 \Rightarrow x_2 = 10^{-2} = \frac{1}{100}\end{eqnarray} - Example 24 - Solve the equation \(\log_4 (\log_3 (\log_2 (x-1))) = 0\)
Solution - \begin{eqnarray}\log_4 (\log_3 (\log_2 (x-1))) &=& 0\\\nonumber \log_3(\log_2(x-1)) &=& 1 \\\nonumber \log_2(x-1) &=& 3\\\nonumber x-1 &=& 8\\\nonumber x &=& 9\end{eqnarray} - Example 25 - Solve the equation \(\log_{0.5} x + \log_8 x = \frac{2}{3}\)
Solution - \begin{eqnarray}\log_{0.5} x + \log_8 x &=& \frac{2}{3}\\\nonumber \log_{0.5} x = \log_\frac{1}{2} x &=& - \log_2 x\\\nonumber \log_8 x = \log_2^3 x &=& \frac{1}{3} \log_2 x\\\nonumber -\log_2 x + \frac{1}{3}\log_2 x = \frac{2}{3}/\cdot 3\\\nonumber -\frac{1}{3}\log_2 x + \log_2 x = 2\\\nonumber -2\log_2 x &=& 2/\cdot (-1)\\\nonumber x &=& 2^{-1} = \frac{1}{2} \end{eqnarray} - Example 26 - Solve the equationa \(\log_2^2 x + 2 \log_2 x = 3\)
Solution - \begin{eqnarray}\log_2^2 x + 2 \log_2 x &=& 3\Rightarrow \log_2 x = t\\\nonumber t^2 + 2t -3 &=& 0\\\nonumber t_{1,2} &=& \frac{-2 \pm \sqrt{4 + 4\cdot 3}}{2} \\\nonumber t_{1,2} &=& \frac{-2 \pm 4}{2} \\\nonumber t_1 &=& \frac{-2 +4 }{2} = 1\\\nonumber t_2 &=& \frac{-2-4}{2} = -3\\\nonumber t_1 = 1 \Rightarrow \log_2x_1 = 1 \Rightarrow x_1 &=& 2\\\nonumber t_2 = -3 \Rightarrow \log_2x_2 = -3 \Rightarrow x_2 &=& 2^{-3} = \frac{1}{8}\end{eqnarray} - Example 27 - Solve the equation \(\log_2(5-x) + 2\log_2 \sqrt{3-x} = 0\)
Solution - \begin{eqnarray}\log_2(5-x) + 2\log_2 \sqrt{3-x} &=& 0\\\nonumber 3-x \geq 0 \Rightarrow x &\leq& 3 \\\nonumber \log_2(5-x)(3-x) &=& 0\\\nonumber x^2 -8x + 14 &=& 0 \\\nonumber x_{1,2} &=& \frac{8 \pm \sqrt{64-4\cdot 14}}{2}\\\nonumber x_{1,2} &=& \frac{8 \pm 2\sqrt{2}}{2} = 4 \pm \sqrt{2}\\\nonumber x_1 &=& 4 + \sqrt{2} \\\nonumber x_2 &=& 4-\sqrt{2}\end{eqnarray} Due to the condition obtained from the expression under the root in the initial form of the equation all values of x variable have to be smaller than 3. The first solution \(x_1 = 4 + \sqrt{2}\) is omitted since the solution is greater than 3. The only solution that remains is \(x_2\) i.e. \(x_2 = 4-\sqrt{2}\) - Example 28 - Solve the equation
\((2\log_2 x - 1)^{-1} + (2\log_2x -3)^{-1} = 4(4\log_2^2x - 8 \log_2 x+3)^{-1}\)
Solution - \begin{eqnarray}(2\log_2 x - 1)^{-1} + (2\log_2x -3)^{-1} &=& 4(4\log_2^2x - 8 \log_2 x+3)^{-1}\\\nonumber \frac{1}{2\log_2 x - 1} + \frac{1}{2\log_2x-3} &=& \frac{1}{4(4\log_2^2x - 8 \log_2 x+3)}\\\nonumber \log_2 x&=& t \\\nonumber \frac{1}{2t-1} + \frac{1}{2t-3} &=& \frac{1}{4(4t^2 - 8t + 3)}\\\nonumber \frac{2t-3+2t-1}{(2t-1)(2t-3)} &=& \frac{1}{4(4t^2 - 8t + 3)}\\\nonumber \frac{4t-4}{4t^2 - 8t +3} &=& \frac{1}{4(4t^2 - 8t + 3)}\\\nonumber 4t-4 &=& \frac{1}{4}/\cdot 4\\\nonumber 16t- 16 &=& 1\\\nonumber 16t &=& 17 /:16 \\\nonumber t &=& \frac{17}{16}\\\nonumber \log_2 x &=& \frac{17}{16}\\\nonumber x &=& 2^{\frac{17}{16}}\\\nonumber x &=& 2\sqrt[16]{2} = 2.08854\end{eqnarray} - Example 29 - Solve the equation \(\log_x 2(\log_x 2 -3) = -2\)
Solution - \begin{eqnarray}\log_x 2(\log_x 2 -3) &=& -2\\ \log_x^2 2 - 3\log_x 2 &=& -2 \\ \log_x 2 = t \\ t^2 -3t +2 &=& 0 \\ t_{1,2} &=& \frac{3\pm\sqrt{9 - 8}}{2} \\ t_{1,2} &=& \frac{3\pm 1}{2}\\ t_1 &=& 2 \\ t_2 &=& 1 \\ \log_x 2 &=& 2 \Rightarrow x^2 = 2 /\sqrt{} \Rightarrow x = \pm \sqrt{2} \\ \log_x 2 &=& 1 \Rightarrow x = 2 \end{eqnarray} - Example 30 - Solve the equation \(\log_2(x-1) + \log_2 x = 1\)
Solution - \begin{eqnarray}\log_2(x-1) + \log_2 x &=& 1\\\nonumber \log_2(x^2-x) &=& 1\\\nonumber x^2 - x &=& 2\\\nonumber x^2 -x -2 &=& 0 \\\nonumber x_{1,2} &=& \frac{1\pm\sqrt{1+4\cdot 2}}{2} \\\nonumber x_{1,2} &=& \frac{1 \pm 3}{2} \\\nonumber x_1 &=& \frac{1+3}{2} = \frac{4}{2} = 2\\\nonumber x_2 &=& \frac{1-3}{2} = -1\end{eqnarray}The negative solution is omitted due to the fact that arguments of log functions must be positive so the only solution is \(x_1 = 2.\) - Example 31 - Solve the equation \(6^{6x} = 66\)
Solution - \begin{eqnarray}6^{6x} &=& 66/\log\\\nonumber 6x\log 6 &=& \log 66 /: 6\log 6 \\\nonumber x &=& \frac{\log 66}{6 \log 6}\end{eqnarray} - Example 32 - Solve the equation \(5^{2x+1} = 2^{5x-1}\)
Solution - \begin{eqnarray}5^{2x+1} &=& 2^{5x-1}/\log \\\nonumber (2x+1) \log 5 &=& 5x\log 2 - \log 2\\\nonumber x(2\log 5 - 5\log 2)&=& -\log 5 - \log 2\\\nonumber x &=& \frac{\log 5 + \log 2}{5\log 2- 2\log 5}\end{eqnarray} - Example 33 - Solve the equation \(7^{x-1} = 13\cdot 10^x\)
Solution - \begin{eqnarray}7^{x-1} &=& 13\cdot 10^x\\\nonumber (x-1) \log 7 &=& \log 13 + x\log10\\\nonumber x(\log 7 - 1) &=& \log 7 + \log 13/:(\log 7 -1) x &=& \frac{\log 7 + \log 13}{log 7 - 1}\end{eqnarray} - Example 34 - Solve the equation \(3^{x-2} = 5\)
Solution - \begin{eqnarray}3^{x-2} &=& 5/\log \\\nonumber (x-2)\log 3 &=& \log 5\\\nonumber x\log 3 - 2\log 3 &=& \log 5\\\nonumber x\log 3 &=& \log 9 + \log 5 /:\log 3\\\nonumber x &=& \frac{\log 45}{\log 3}\end{eqnarray} - Example 35 - Solve the equation \(\log_{\frac{1}{7}} x = -3\)
Solution - \begin{eqnarray}\log_{\frac{1}{7}} x &=& -3\\\nonumber -\log_7 x &=& -3 /\cdot(-1) \\\nonumber x&=& 7^3 = 343\end{eqnarray} - Example 36 - Solve the equation \(\log_3 x = 3\)
Solution - \begin{eqnarray}\log_3 x &=& 3\\\nonumber x &=& 3^3 = 27\end{eqnarray} - Example 37 - Solve the equation \(\log x = 4\)
Solution - \begin{eqnarray}\log x &=& 4 \\\nonumber x &=& 10^4 = 10000\end{eqnarray} - Example 38 - Solve the equation \(\log_\sqrt{2} (2x-4) = 4\)
Solution - \begin{eqnarray}\log_\sqrt{2} (2x-4) &=& 4\\\nonumber 2\log_2(2x-4) &=& 4/:2 \\\nonumber 2x - 4 &=& 4\\\nonumber 2x &=& 8/:2\\\nonumber x &=& 4\end{eqnarray} - Example 39 - Solve the equation \(\log_x 3 = 3\)
Solution - \begin{eqnarray}\log_x 3 &=& 3\\\nonumber 3 &=& x^3 \\\nonumber x &=& \sqrt[3]{3}\end{eqnarray} - Example 40 - Solve the equation \(\log_x 81 = 2\)
Solution - \begin{eqnarray}\log_x 81 &=& 2\\\nonumber \log_x 3^4 &=& 2\\\nonumber x^2 &=& 3^4 /^\sqrt{} \\\nonumber x_{1,2} &=& \pm 9\end{eqnarray} - Example 41 - Solve the equation \(\log_x 25 = -2\)
Solution - \begin{eqnarray}\log_x 25 &=& -2\\\nonumber \frac{1}{x^2} &=& 25 \\\nonumber x^2 &=& \frac{1}{25}/^\sqrt{}\\\nonumber x_{1,2} &=& \pm \frac{1}{5}\end{eqnarray} - Example 42 - Solve the equation \(\log_{x+1} 16 = 2\)
Solution - \begin{eqnarray}\log_{x+1} 16 &=& 2\\\nonumber (x+1)^2 &=& 16 \\\nonumber x^2 + 2x -15 &=& 0 \\\nonumber x_{1,2} &=& \frac{-2 \pm \sqrt{4 + 4\cdot 15}}{2} \\\nonumber x_{1,2} &=& \frac{-2 \pm 8}{2}\\\nonumber x_1 &=& \frac{-2+8}{2} = 3\\\nonumber x_2 &=& \frac{-2-8}{2} = -5\end{eqnarray} The \(x_2 = -5\) can not be solution since the value is negative. - Example 43 - Solve the equation \(\log \frac{1}{x^2} + \log \frac{1}{x} + \log x + \log x^2 + \log x^3 = 6\)
Solution - \begin{eqnarray}\log \frac{1}{x^2} + \log \frac{1}{x} + \log x + \log x^2 + \log x^3 &=& 6\\\nonumber -2 \log x - \log x + \log x + 2\log x + 3 \log x &=& 6\\\nonumber x &=& 10^2 \\\nonumber x &=& 100\end{eqnarray} - Example 44 - Solve the equation \(\log_x 16 -\frac{1}{2} = \log_x 8\)
Solution - \begin{eqnarray}\log_x 16 -\frac{1}{2} &=& \log_x 8\\\nonumber 4 \log_x 2 - 3 \log_x 2 &=& \frac{1}{2}\\\nonumber \log_x 2 &=& \frac{1}{2} \\\nonumber \sqrt{x} &=& 2/^2 \\\nonumber x &=& 4\end{eqnarray} - Example 45 - Solve the equation \(\ln 2x + \ln x^2 - \ln \sqrt[3]{x} = 1 + \ln 2 - \ln x^{-3}\)
Solution - \begin{eqnarray}\ln 2x + \ln x^2 - \ln \sqrt[3]{x} &=& 1 + \ln 2 - \ln x^{-3}\\\nonumber \ln 2 + \ln x + 2 \ln x - \frac{1}{3}\ln x &=& 1 + \ln 2 +3 \ln x \\\nonumber -\frac{1}{3} \ln x &=& 1/\cdot (-3) \\\nonumber \ln x &=& -3 \\\nonumber x &=& e^{-3}\\\nonumber x&=& \frac{1}{e^3}\end{eqnarray} - Example 46 - Solve the equation \(\frac{\ln x^\pi -1}{\ln x + 1} = 1\)
Solution - \begin{eqnarray}\frac{\ln x^\pi -1}{\ln x + 1} &=& 1/\cdot(\ln x+ 1)\\\nonumber \pi \ln x - \ln x &=& 2\\\nonumber \ln x(\pi -1) &=& 2 \\\nonumber \ln x &=& \frac{2}{\pi - 1}\\\nonumber x &=& e^{\frac{2}{\pi-1}}\end{eqnarray} - Example 47 - \(\log_2(9-2^x) = 3-x\)
Solution - \begin{eqnarray}\log_2(9-2^x) &=& 3-x\\\nonumber 9-2^x &=& 2^{3-x}\\\nonumber 9-2^x - 2^3 \cdot 2^{-x} &=& 0 /\cdot(-2^x)\\\nonumber 2^{2x} - 9\cdot 2^x + 8 &=& 0 \Rightarrow 2^x = t \\\nonumber t^2 - 9t + 8 &=& 0\\\nonumber t_{1,2} &=& \frac{9 \pm \sqrt{81-4\cdot 8}}{2}\\\nonumber t_{1,2} &=& \frac{9\pm 7}{2}\\\nonumber t_1 &=& \frac{9+7}{2} = 8 \\\nonumber t_2 &=& \frac{9-7}{2} = 1\\\nonumber 2^{x_1} &=& 2^3 \\\nonumber x_1 &=& 3 \\\nonumber 2^{x_2} &=& 2 \\\nonumber x_2 &=& 1 \end{eqnarray} - Example 48 - Solve the equation \(\left(\frac{4}{9}\right)^x \left(\frac{27}{8}\right)^{x-1} = \frac{\log 4}{\log 8}\)
Solution - \begin{eqnarray}\left(\frac{4}{9}\right)^x \left(\frac{27}{8}\right)^{x-1} &=& \frac{\log 4}{\log 8}\\\nonumber \left(\frac{2}{3}\right)^{2x}\left(\frac{3}{2}\right)^{3x-3} &=& \frac{2}{3}\\\nonumber \left(\frac{3}{2}\right)^{-2x} \left(\frac{3}{2}\right)^{3x-3} &=& \left(\frac{3}{2}\right)^{-1}\\\nonumber -2x +3x -3 &=& -1 \\\nonumber x-3 &=& -1 \\\nonumber x &=& 2\end{eqnarray} - Example 49 - Solve the equation \(7^{\log x} + 13 \cdot 7 ^{\log x-1} - 5^{\log x +1} -3 \cdot 5^{\log x -1} = 0\)
Solution - \begin{eqnarray}7^{\log x} + 13 \cdot 7 ^{\log x-1} - 5^{\log x +1} -3 \cdot 5^{\log x -1} &=& 0/:5^{\log x}\\\nonumber \left(\frac{7}{5}\right)^{\log x} + 13 \cdot \left(\frac{7}{5}\right)^{\log x} \cdot \frac{1}{7} - 5 - \frac{3}{5} &=& 0\\\nonumber \left(\frac{7}{5}\right)^{\log x} + \left(\frac{13}{7}\right)\left(\frac{7}{5}\right)^{\log x} &=& \frac{25 + 3}{5}\\\nonumber \left(\frac{7+13}{5}\right) \left(\frac{7}{5}\right)^{\log x} &=& \frac{28}{5}\\\nonumber \frac{20}{7}\left(\frac{7}{5}\right)^{\log x} &=& \frac{28}{5}/\cdot\frac{7}{5}\\\nonumber \left(\frac{7}{5}\right)^{\log x} = \frac{49}{25}\\\nonumber \left(\frac{7}{5}\right)^{\log x} &=& \left(\right)^2 \\\nonumber \log x &=& 2 \\\nonumber x &=& 10^2 = 100 \end{eqnarray} - Example 50 - Solve the equation \(\log_4(\log_3(\log_2x)) = \frac{1}{2}\)
Solution - \begin{eqnarray}\log_4(\log_3(\log_2x)) &=& \frac{1}{2}\\\nonumber \log_3(\log_2x)&=& 2 \\\nonumber \log_2 &=& 9 \\\nonumber x &=& 2^9 \\\nonumber x &=& 512\end{eqnarray} - Example 51 - Solve the equation \(\log_x 10 + \log_x^2 10 = 2\)
Solution - \begin{eqnarray}\log_x 10 + \log_x^2 10 &=& 2\\\nonumber \log_x 10 +\frac{1}{2}\log_x 10 &=& 2 \\\nonumber \frac{3}{2}\log_x 10 &=& 2 /\cdot \frac{2}{3}\\\nonumber \log_x 10 &=& \frac{4}{3}\\\nonumber x^{\frac{4}{3}} &=& 10 \\\nonumber x &=& \sqrt[4]{1000}\end{eqnarray} - Example 52 - Solve the equation \(\log_2 x + \log_4 x + \log_{16} x = 7\)
Solution - \begin{eqnarray}\log_2 x + \log_4 x + \log_{16} x &=& 7\\\nonumber \log_2 x + \frac{1}{2}\log_2 x + \frac{1}{4}\log_2 x &=& 7 \\\nonumber \frac{4 +2 +1}{4} \log_2 x &=& 7 \\\nonumber \frac{7}{4}\log_2 x &=& 7 /\cdot 4 \\\nonumber \log_2 x &=& 4 \\\nonumber x&=& 2^4 \\\nonumber x &=& 16\end{eqnarray} - Example 53 - Solve the equation \(x^{1+\log x} = 100\)
Solution - \begin{eqnarray}x^{1+\log x} &=& 100/\log\\\nonumber (1+\log x) \log x &=& 2 \\\nonumber \log^2 x + \log x 2 &=& 0 \\\nonumber \log x &=& t \\\nonumber t^2 + t -2 &=& 0 \\\nonumber t_{1,2} &=& \frac{-1 \pm \sqrt{1+4\cdot 2}}{2} \\\nonumber t_{1,2} &=& \frac{-1\pm 3}{2}\\\nonumber t_1 &=& 1 \\\nonumber t_2 &=& -2\\\nonumber \log x_1 = 1 \Rightarrow x_1 &=& 10 \\\nonumber \log x_2 = -2 \Rightarrow x_2 &=& 10^{-2} = \frac{1}{100} \end{eqnarray} - Example 54 - Solve the equation \(\log_x(9x^2) \log_3^2 x= 4\)
Solution \begin{eqnarray} \log_x(9x^2) \log_3^2 x&=& 4\\ \log_x(9)\log_3^2 x + \log_xx^2 \log_3^2 x &=& 4\\ 2\log_x 3 \log_3^2x + 2\log_3^2x &=& 4/:2 \\ \log_x 3 \log_3^2x + \log_3^2x &=& 2 \\ \frac{\log_3^2x}{\log_3 x} + \log_3^2x &=& 2 \\ \log_3 x + \log_3^2 x &=& 2 \\ \log_3 x &=& t \\ t^2 + t -2 &=& 0 \\ t_{1,2} &=& \frac{-1 \pm \sqrt{1+8}}{2}\\ t_{1,2} &=& \frac{-1 \pm 3}{2}\\ t_1 &=& \frac{2}{2} = 1 \\ t_2 &=& -\frac{4}{2} = -2 \\ \log_3 x_1 &=& 1 \Rightarrow x_1 = 3 \\ \log_3 x_2 &=& -2 \Rightarrow x_2 = \frac{1}{9} \end{eqnarray} - Example 55 - Solve the equation \(\log_2 (4^x + 1) = x+ \log_2 (2^{x+3} -6)\)
Solution - \begin{eqnarray}\log_2 (4^x + 1) &=& x+ \log_2 (2^{x+3} -6)\\\nonumber \log_2 \frac{2^{2x} +1 }{2^{x+3}-6} &=& 1 \\\nonumber 2^{2x} + 1 &=& 2^x(2^{x+3} -6)\\\nonumber 2^{2x} + 1 &=& 2^{2x} 2^3 - 3\cdot 2^{x+1}\\\nonumber -7\cdot 2^{2x} + 3\cdot 2^{x+1} + 1 &=& 0/\cdot (-1) \\\nonumber 7 \cdot 2^{2x} -6\cdot 2^{x} -1 &=& 0\Rightarrow 2^{x} = t \\\nonumber 7 t^2 -6t -1 &=& 0\\\nonumber t_{1,2} &=& \frac{6 \pm \sqrt{36 + 4\cdot 7}}{2\cdot 7}\\\nonumber t_{1,2} &=& \frac{6\pm 8}{14}\\\nonumber t_1 &=& 1 \\\nonumber t_2 &=& -\frac{1}{7}\\\nonumber 2^x = 1 \Rightarrow X_1 &=& 0 \end{eqnarray} - Example 56 - Solve the equation \(2\cdot 9^{\log x} - 8 \cdot 3^{\log x} = 90\)
Solution - \begin{eqnarray}2\cdot 9^{\log x} - 8 \cdot 3^{\log x} &=& 90\\\nonumber 2 \cdot 3^{2\log x} - 8 \cdot 3^{\log x} -90 &=& 0\\\nonumber 3^{2\log x} - 4 \cdot 3^{\log x} - 45 &=& 0 \Rightarrow 3^{\log x} = t\\\nonumber t^2 - 4t - 45 &=& 0 \\\nonumber t_{1,2} &=& \frac{4\pm \sqrt{16 + 4\cdot 45}}{2}\\\nonumber t_{1,2} &=& \frac{4 \pm 14}{2} \\\nonumber t_1 &=& 9 \\\nonumber t_2 &=& -5 \\\nonumber 3^{\log x} &=& 3^2 \Rightarrow \log x_1 = 2 \Rightarrow x_1 = 100\\\nonumber \end{eqnarray} - Example 57 - Solve the system of equations \(\begin{cases}\log(x^2+y^2) = 1+ \log 13 \\ \log(x+y) - \log(x-y) = 3\log 2\end{cases}\)
Solution - \begin{eqnarray}\log(x+y) - \log(x-y) &=& 3\log 2 \\\nonumber \frac{x+y}{x-y} &=& 8 /\cdot(x-y)\\\nonumber x+y &=& 8x-8y \\\nonumber -7x &=& -9y\\\nonumber x &=& \frac{9}{7}y\end{eqnarray} \begin{eqnarray}\log(x^2+y^2) &=& 1+ \log 13\\\nonumber \log\left(\frac{81}{49}y^2 + y^2\right) &=& \log 10 + \log 13\\\nonumber \log\left(\frac{81 + 49}{49}y^2\right) &=& \log 130 \\\nonumber \frac{130}{49}y^2 &=& 130/\cdot 49\\\nonumber y^2 &=& 49 \Rightarrow y_{1,2} =\pm 7 \end{eqnarray} - Example 58 - Solve the system of equations \(\begin{cases}\log_2(x-y) = 4 - \log_2(x-y)\\ \frac{\log\frac{x}{4}}{\log\frac{y}{3}} = 1\end{cases}\)
Solution - \begin{eqnarray}\frac{\log\frac{x}{4}}{\log\frac{y}{3}} &=& 1/\cdot\log\frac{y}{3} \\\nonumber \frac{x}{4} &=& \frac{y}{3} \\\nonumber x &=& \frac{4}{3}y\end{eqnarray} \begin{eqnarray}\log_2(x-y)^2 &=& 4 \\\nonumber \log_2(x^2 -2xy + y^2) &=& 4\\\nonumber \frac{16}{9} y^2 - 2\frac{4}{3}y^2 + y^2 &=& 2^4\\\nonumber \frac{1}{9}y^2 &=& 16/\cdot 9\\\nonumber y&=& \sqrt{144} = 12 \\\nonumber x &=& \frac{4}{3}12 \\\nonumber x &=& 16\end{eqnarray}
- Home
- Solvers
- Numbers and Algebra
- Functions
- Geometry
- Calculus
- Numerical Methods
- Website Information
How to solve logarithmic equations?
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment