# Exercises - Exponential and Logarithmic Equations

1. Solve for $x$:

1. $\displaystyle{7^{2x-1} = 1}$

2. $\displaystyle{e^{x^2-1} - 1 = 0}$

3. $\displaystyle{4^{-x+1} = \frac{1}{32}}$

4. $\displaystyle{2^{5x} = 3}$

5. $\displaystyle{e^{2x+1} = 7}$

6. $\displaystyle{27^x = \frac{9^{2x-1}}{3^{2x}}}$

7. $\displaystyle{3^{2x} - 2(3^x) - 3 = 0}$

8. $\displaystyle{2^x + 2^{-x} = 2}$

2. Solve for $x$:

1. $\displaystyle{\log_{10} (2x+50) = 2}$

2. $\displaystyle{\log_2 x + \log_2 (x-2) = 3}$

3. $\displaystyle{\log_3 (7-x) = \log_3 (1-x) + 1}$

4. $\displaystyle{(\log_{10} x)^2 + \log_{10} x = 2}$

5. $\displaystyle{\frac{\log_3 16}{2\,\log_3 x} = 2}$

6. $\displaystyle{\log_3 (\log_2 x) = 1}$

7. $\displaystyle{9^{\log_3 x} = 4}$

8. $\displaystyle{e^{-\ln x} = x}$

3. Simplify:

1. $\displaystyle{\log_{16} \frac{1}{\sqrt[5]{64}}}$   $\ans{\displaystyle{-\frac{3}{10}}}$

2. $\displaystyle{e^{4\ln a \, + \, \frac{1}{2} \ln b}}$   $\ans{\displaystyle{a^4 \sqrt{b}}}$

3. $\displaystyle{\frac{1}{\ln e^{-a}}}$   $\ans{\displaystyle{-\frac{1}{a}}}$

4. $\displaystyle{\frac{1}{2} \log_4 16}$   $\ans{\displaystyle{1}}$

5. $\displaystyle{\frac{1}{2} \log_3 9 - \frac{3}{4} \log_3 16 + \frac{3}{2} \log_3 4}$   $\ans{\displaystyle{1}}$

6. $\displaystyle{e^{-\ln 5}}$   $\ans{\displaystyle{\frac{1}{5}}}$

7. $\displaystyle{e^{-\frac{1}{2} \ln \frac{1}{16} \, - \, \frac{2}{3} \ln 27}}$   $\ans{\displaystyle{\frac{4}{9}}}$

8. $\displaystyle{\log_3 \sqrt[5]{\frac{1}{81}}}$   $\ans{\displaystyle{-\frac{4}{5}}}$

9. $\displaystyle{\frac{1}{2} \log_3 64 - 2 \log_3 2 + \frac{1}{2} \log_3 \frac{1}{4}}$   $\ans{\displaystyle{0}}$

10. $\displaystyle{\log_{1/2} 64}$   $\ans{\displaystyle{-6}}$

11. $\displaystyle{8^{\log_8 30}}$   $\ans{\displaystyle{30}}$

12. $\displaystyle{\log_{1/3} \sqrt[9]{27^2}}$   $\ans{\displaystyle{-\frac{2}{3}}}$

13. $\displaystyle{2 \log_5 4 - \frac{1}{2} \log_5 64 - \log_5 2}$   $\ans{\displaystyle{0}}$

14. $\displaystyle{\ln \left[e^{\ln 5 \, - \, \frac{1}{2} \ln 25} \right] + \ln e^4 - \ln \left[ e^{\ln e} \right]}$   $\ans{\displaystyle{3}}$

4. Solve the following exponential equations:

1. $\displaystyle{5^{x-2} = 1}$   $\ans{\displaystyle{2}}$

2. $\displaystyle{10^{-2x} = \frac{1}{10,000}}$   $\ans{\displaystyle{2}}$

3. $\displaystyle{2^x + 2^{-x} = 3}$   $\ans{\displaystyle{\log_2 \left( \frac{3 \pm \sqrt{5}}{2} \right)}}$

4. $\displaystyle{2^{x^2}=8^{2x-3}}$   $\ans{\displaystyle{3}}$

5. $\displaystyle{\frac{1}{4}(10^{-2x}) - 25(10^x) = 0}$   $\ans{\displaystyle{-\frac{2}{3}}}$

6. $\displaystyle{4^{\log_2 x} = 9}$   $\ans{\displaystyle{3 \textrm{ only}}}$

7. $\displaystyle{e^{-\ln x} = 5}$   $\ans{\displaystyle{\frac{1}{5}}}$

8. $\displaystyle{\left( \frac{1}{2} \right)^{2-x} = 8 \left( 2^{x-1} \right)^3}$   $\ans{\displaystyle{-1}}$

9. $\displaystyle{6^{2x} - 7\,(6^x) + 6 = 0}$   $\ans{\displaystyle{0,1}}$

10. $\displaystyle{\frac{1}{3} = \left( 2^{\,|x| - 2} - 1 \right)^{-1}}$   $\ans{\displaystyle{4,-4}}$

5. Solve the following logarithmic equations:

1. $\displaystyle{\log_{10} \frac{1}{x^2} = 2}$   $\ans{\displaystyle{\pm \frac{1}{10}}}$

2. $\displaystyle{\log_3 \sqrt{x^2+17} = 2}$   $\ans{\displaystyle{\pm 8}}$

3. $\displaystyle{\log_2 (\log_3 x) = 2}$   $\ans{\displaystyle{81}}$

4. $\displaystyle{\log_4 x^2 = (\log_4 x)^2}$   $\ans{\displaystyle{1,16}}$

5. $\displaystyle{\log_6 2x - \log_6 (x+1) = 0}$   $\ans{\displaystyle{1}}$

6. $\displaystyle{2\ln x = 1}$   $\ans{\displaystyle{\sqrt{e}}}$

7. $\displaystyle{\frac{\log_2 8^x}{\log_2 \frac{1}{4}} = \frac{1}{2}}$   $\ans{\displaystyle{-\frac{1}{3}}}$

8. $\displaystyle{\log_9 \sqrt{10x+5} - \frac{1}{2} = \log_9 \sqrt{x+1}}$   $\ans{\displaystyle{4}}$

6. Solve the following equations for $x$:

1. $\displaystyle{\log_x 2 = -\frac{1}{4}}$   $\ans{\displaystyle{\frac{1}{16}}}$

2. $\displaystyle{\log_5 x = -3}$   $\ans{\displaystyle{\frac{1}{125}}}$

3. $\displaystyle{\log_{1/2} 8 = x}$   $\ans{\displaystyle{-3}}$

4. $\displaystyle{\log_x 4 = \frac{2}{3}}$   $\ans{\displaystyle{8}}$

5. $\displaystyle{6^x = 5}$   $\ans{\displaystyle{\log_6 5}}$

6. $\displaystyle{\log_x b = 3}$   $\ans{\displaystyle{\sqrt[3]{b}}}$

7. $\displaystyle{\ln x = -1}$   $\ans{\displaystyle{\frac{1}{e}}}$

8. $\displaystyle{2^{\ln x} = 4}$   $\ans{\displaystyle{e^2}}$

9. $\displaystyle{\log_x 8 = -\frac{3}{5}}$   $\ans{\displaystyle{\frac{1}{32}}}$

10. $\displaystyle{e^{\frac{1}{2} \ln (x+1)} = 3}$   $\ans{\displaystyle{8}}$

11. $\displaystyle{\ln x = e}$   $\ans{\displaystyle{e^e}}$

12. $\displaystyle{\log_2 2x + \log_2 (x+\frac{3}{2}) = 1}$   $\ans{\displaystyle{\frac{1}{2} \textrm{ only}}}$

13. $\displaystyle{\ln \sqrt{e^x} = -3}$   $\ans{\displaystyle{-6}}$

14. $\displaystyle{2 \log_2 (x+2) - \log_2 (x^2-4) = 3}$   $\ans{\displaystyle{\frac{18}{7} \textrm{ only}}}$

15. $\displaystyle{\log_4 \frac{2}{3x} = 3}$   $\ans{\displaystyle{\frac{1}{96}}}$

16. $\displaystyle{\log_x 8 = -\frac{3}{4}}$   $\ans{\displaystyle{\frac{1}{16}}}$

17. $\displaystyle{\frac{1}{2} \log_3 (x^2-19) = 2}$   $\ans{\displaystyle{\pm 10}}$

18. $\displaystyle{2^{x^2+4x} = \frac{1}{8}}$   $\ans{\displaystyle{-3,-1}}$