Practice 1 Solutions

1. ${\left(\ln e^2\right)}^6$

$2^6$

64

2. $e^{\left(\ln 2–\ln 3\right)}$

$e^{\left(\ln \frac{2}{3}\right)}$

$\frac{2}{3}$

4. $e^{\left(\ln 5+\ln 6\right)}$

$$\begin{gathered} e^{\left(\ln 5·6\right)} \\ {e}^{\ln 30} \end{gathered}$$

30

5. $3 \ln x+2 \ln y–4\left(\ln a+\ln b\right)$

$$\begin{gathered} \ln x^3+\ln y^2–4 \ln \left(ab\right) \\ \ln x^{3}y{\left[\right]}^2–\ln {\left(ab\right)}^4 \end{gathered}$$

$\ln \frac{x^3{y}^2}{{\left(ab\right)}^4}$

6. $\frac{1}{3} \ln 12–\ln 7$

$$\begin{gathered} \ln {12}^{\frac{1}{3}}–\ln 7 \\ \ln \sqrt[3]{12}–\ln 7 \end{gathered}$$

$\ln \frac{\sqrt[3]{12}}{7}$

7. $\ln \frac{x{y}^4}{z^2}$

$$\begin{gathered} \ln \left(x{y}^4\right)– \ln z^2 \\ \ln x+ \ln y^4– \ln z^2 \end{gathered}$$

$\ln x+ 4 \ln y– 2 \ln z$

8. $\ln \frac{\sqrt[4]{x^3}}{y{z}^5}$

$$\begin{gathered} \ln \frac{x^{{\displaystyle \frac{3}{4}}}}{y{z}^5} \\ \ln x^{\frac{3}{4}}– \ln y{z}^5 \\ \ln x^{\frac{3}{4}}–\ln y – \ln z^5 \end{gathered}$$

$\frac{3}{4} \ln x– \ln y– 5 \ln z$

9. $6^{\frac{x}{4}+2}=51$

$\begin{array}{rcl}\ln 6^{\frac{x}{4}+2}& =& \ln 51\\ \left(\frac{x}{4}+2\right) \ln 6& =& \ln 51\\ \frac{x}{4}+2& =& \frac{\ln 51}{\ln 6}\\ & & \\ \frac{x}{4}& =& \frac{\ln 51}{\ln 6}–2\end{array}$

$x=\frac{4 \ln 51}{\ln 6}–8\approx 0.7776$

10. ${17}^{x+1}=34$

$\begin{array}{rcl}\ln {17}^{x+1}& =& \ln 34\\ \left(x+1\right) \ln 17& =& \ln 34\\ x+1& =& \frac{\ln 34}{\ln 17}\end{array}$

$x=\frac{\ln 34}{\ln 17}–1\approx 0.2447$

11. $7^{5x+4}=56$

$\begin{array}{rcl}\ln 7^{5x+4}& =& \ln 56\\ \left(5x+4\right)\ln 7& =& \ln 56\\ 5x+4& =& \frac{\ln 56}{\ln 7}\\ & & \\ 5x& =& \frac{\ln 56}{\ln 7}–4\end{array}$

$x=\frac{\ln 56}{5 \ln 7}–\frac{4}{5}\approx –0.3863$

12. $2^x=83$

$\begin{array}{rcl}\ln 2^x& =& \ln 83\\ x \ln 2& =& \ln 83\end{array}$

$x=\frac{\ln 83}{\ln 2}\approx 6.3750$

13. $\frac{1}{2}=\ln \left(4x–3\right)$

$\begin{array}{rcl}{e}^{\frac{1}{2}}& =& e^{\ln \left(4x–3\right)}\\ e^{\frac{1}{2}}& =& 4x–3\\ \frac{e^{{\displaystyle \frac{1}{2}}}+3}{4}& =& \frac{4x}{4}\\ x& =& \frac{e^{{\displaystyle \frac{1}{2}}}+3}{4}\end{array}$

x = 1.1622

14. $\ln {\left(5x\right)}^{\frac{2}{3}}=8$

$\begin{array}{rcl}\frac{2}{3}\ln \left(5x\right)& =& 8\\ \left(\frac{3}{2}\right)\frac{2}{3}\ln \left(5x\right)& =& 8\left(\frac{3}{2}\right)\\ \ln 5x& =& 12\\ e^{\ln 5x}& =& e^{12}\\ \frac{5x}{5}& =& \frac{e^{12}}{5}\\ x& =& \frac{e^{12}}{5}\end{array}$

x = 32,550.9583

16. $\frac{1}{3}\ln \left(x+6\right)=\ln 2$

$\begin{array}{rcl}\ln {\left(x+6\right)}^{\frac{1}{3}}& =& \ln 2\\ {\left(x+6\right)}^{\frac{1}{3}}& =& 2\\ {\left({\left(x+6\right)}^{\frac{1}{3}}\right)}^3& =& 2^3\\ x+6& =& 8\end{array}$

x = 2