## Exercises - The 'Natural' Base

1. Find the values of the following:

1. $\ln \sqrt{e}$

2. $\ln \sqrt[3]{e^2}$

3. $\ln (e^2 \cdot e^3)$

4. $\ln (e^2)^3$

5. $\ln \frac{1}{\sqrt[3]{e^2}}$

6. $\ln \left(\frac{e^{3/2}}{e^2 \sqrt{e}} \right)$

7. $\ln \frac{\sqrt{e^3}}{e}$

8. $e^{-\ln 3}$

9. $e^{\frac{1}{2}\ln \frac{1}{16} - \frac{2}{3} \ln 27} - \ln e^{\frac{5}{4}}$

1. $\frac{1}{2}$

2. $\frac{2}{3}$

3. $5$

4. $6$

5. $-\frac{2}{3}$

6. $-1$

7. $\frac{1}{2}$

8. $\frac{1}{3}$

9. $\frac{-11}{9}$

2. If the given statement is in exponential form, write it in logarithmic form. If instead it is in logarithmic form, write it in exponential form:

1. $e^y = 3$

2. $e^5 = x$

3. $\ln x = 3$

4. $\ln 3x = -2$

5. $\ln e^2 = 2$

1. $\ln 3 = y$

2. $\ln x = 5$

3. $e^3 = x$

4. $e^{-2} = 3x$

5. $e^2 = e^2$

3. Simplify and write as one logarithm:

1. $3\ln 5 - \frac{1}{2}\ln 4 + \ln 8$

2. $\ln(x^3+8)-\ln(x^2-2x+4)-2\ln(x+2)$

1. $\ln 500$

2. $-\ln (x+2)$

4. Solve for the unknown

1. $\ln x = 3$

2. $\ln \sqrt{e} = x$

3. $\ln e^{2x} = -\frac{1}{2}$

4. $\ln e^4 = x^2$

5. $(\ln x)^2 = \ln x^2$

6. $e^{-\frac{1}{2} \ln x} = 4$

7. $\ln x^3 = \frac{1}{e}$

8. $\ln x = \ln 1 + \ln 2 + \ln 3 + \ln 4$

9. $\log_2 (x-7) + \log_2 x = \ln e^3$

10. $e^{\ln 2x} - \ln e^{3x} = -3$

1. $e^3$

2. $\frac{1}{2}$

3. $-\frac{1}{4}$

4. $\pm 2$

5. $1,e^2$

6. $\frac{1}{16}$

7. $e^{\frac{1}{3e}}$

8. $24$

9. $8$ only

10. $3$

5. Solve for $x$:

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

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

3. $e^{-\ln x} = x$

4. $e^{-\ln x} = 5$

5. $2 \ln x = 1$

6. $\ln x = -1$

7. $2^{\ln x} = 4$

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

9. $\ln x = e$

10. $\ln \sqrt{e^x}=-3$

1. $x=\pm 1$

2. $x = \frac{\ln 7 - 1}{2}$

3. $x = 1$

4. $x = \frac{1}{5}$

5. $\sqrt{e}$

6. $\frac{1}{e}$

7. $e^2$

8. $8$

9. $e^e$

10. $-6$