Practice 2 Solutions

1. $e^{\left(\ln 3 + \ln 9\right)}$

$$\begin{gathered} e^{\left(\ln 3·9\right)} \\ {e}^{\ln 27} \end{gathered}$$

27

2. $\ln e^5– \ln e^2$

5 – 2

3

3. $e^{\left(\ln 8– \ln 9\right)}$

$e^{ \ln \frac{8}{9}}$

$\frac{8}{9}$

4. ${\left(\ln e^5\right)}^2$

$5^2$

25

5. $\frac{1}{3} \ln x+\frac{2}{3} \ln y– \ln 7$

$$\begin{gathered} \ln x^{\frac{1}{3}}+ \ln y^{\frac{2}{3}}– \ln 7 \\ \ln \left(x^{\frac{1}{3}}{y}^{\frac{2}{3}}\right)– \ln 7 \\ \ln \sqrt[3]{x{y}^2}–\ln 7 \end{gathered}$$

$\ln \frac{\sqrt[3]{x{y}^2}}{7}$

6. $3 \ln 2 + \ln 6 – \ln 5$

$$\begin{gathered} \ln 2^3+ \ln 6 – \ln 5 \\ \ln 8+ \ln 6 – \ln 5 \\ \ln 48 – \ln 5 \end{gathered}$$

$\ln \frac{48}{5}$

7. $\ln \sqrt[5]{x^2{y}^3}$

$$\begin{gathered} \ln {\left(x^2{y}^3\right)}^{\frac{1}{5}} \\ \ln \left(x^{\frac{2}{5}}{y}^{\frac{3}{5}}\right) \\ \ln x^{\frac{2}{5}}+ \ln y^{\frac{3}{5}} \end{gathered}$$

$\frac{2}{5} \ln x+\frac{ 3}{5} \ln y$

8. $\ln \frac{2y{\left(x+3\right)}^2}{z}$

$\ln 2 +\ln y + \ln {\left(x+3\right)}^2– \ln z$

$\ln 2 + \ln y+ 2 \ln \left(x+3\right)– \ln z$

9. ${215}^{x+4}=37$

$\begin{array}{rcl}\ln {215}^{x+4}& =& \ln 37\\ \left(x+4\right) \ln 215& =& \ln 37\\ x+4& =& \frac{\ln 37}{\ln 215}\end{array}$

$x=\frac{\ln 37}{\ln 215}–4\approx –3.3277$

10. ${71}^{0.3x}=19$

$\begin{array}{rcl}\ln {71}^{0.3x}& =& \ln 19\\ 0.3x \ln 71& =& \ln 19\\ 0.3x& =& \frac{\ln 19}{\ln 71}\end{array}$

$x=\frac{\ln 19}{0.3 \ln 71}\approx 2.3025$

11. $5^x=12$

$\begin{array}{rcl}\ln 5^x& =& \ln 12\\ x \ln 5& =& \ln 12\end{array}$

$x=\frac{\ln 12}{\ln 5}\approx 1.5440$

12. $9^{\frac{2}{3}x–1}=15$

$\begin{array}{rcl}\ln 9^{\frac{2}{3}x–1}& =& \ln 15\\ \left(\frac{2}{3}x–1\right) \ln 9& =& \ln 15\\ \frac{2}{3}x–1& =& \frac{\ln 15}{\ln 9}\\ \frac{2}{3}x& =& \frac{\ln 15}{\ln 9}+1\end{array}$

$x = \frac{3 \ln 15}{2 \ln 9}+\frac{3}{2}\approx 3.3487$

13. $\ln \left(2x+1\right)–\ln x=3$

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

x = 0.0553

14. $\ln \left(2x–5\right)=6$

$\begin{array}{rcl}{e}^{\ln \left(2x–5\right)}& =& e^6\\ 2x–5& =& e^6\\ \frac{2x}{2}& =& \frac{e^6+5}{2}\\ x& =& \frac{e^6+5}{2}\end{array}$

x = 204.2144

15. $7^{x–3}=21$

$\begin{array}{rcl}\ln 7^{x–3}& =& \ln 21\\ \left(x–3\right)\ln 7& =& \ln 21\\ x–3& =& \frac{\ln 21}{\ln 7}\\ x& =& \frac{\ln 21}{\ln 7}+3\end{array}$

x = 4.5646

16. $\ln 2–\ln \left(x–1\right)=3$

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

x = 1.0996