Practice 1 Solutions

1. $\log 154$

$$\begin{gathered} {10}^x=154 \\ \begin{array}{cc}{10}^2=100& {10}^3=1000\end{array} \end{gathered}$$

Between 2 and 3

2. $\log 4$

$$\begin{gathered} {10}^x=4 \\ \begin{array}{cc}{10}^0=1& {10}^1=10\end{array} \end{gathered}$$

Between 0 and 1

3. $\log 6438$

$$\begin{gathered} {10}^x=6438 \\ \begin{array}{cc}{10}^3=1000 & {10}^4=10000\end{array} \end{gathered}$$

Between 3 and 4

4. $\log \frac{1}{25}$

$$\begin{gathered} {10}^x=\frac{1}{25} \\ \begin{array}{cc}{10}^{–1}=\frac{1}{10}& {10}^{–2}=\frac{1}{100}\end{array} \end{gathered}$$

Between –2 and –1

5. $\log 108$

$\begin{array}{rcl}& & \begin{array}{c}\log 108=\log \left(2^2·3^3\right)\\ =\log 2^2+\log 3^3\end{array}\end{array}$

1. $\begin{array}{c}=2 \log 2+3 \log 3\\ =2\left(X\right)+3\left(Y\right)\\ =2X+3Y\end{array}$
2. $\begin{array}{c}\approx 2(0.301)+3(0.4771)\\ \approx 2.0333\end{array}$

1. $2X+3Y$
2. 2.0333

6. $\log 96$

$\begin{array}{rcl}& & \begin{array}{c}\log 96=\log (2^5·3)\\ =\log 2^5+\log 3\end{array}\end{array}$

1. $\begin{array}{c}=5 \log 2+ \log 3\\ =5\left(X\right)+\left(Y\right)\\ =5X+Y\end{array}$
2. $\begin{array}{c}\approx 5(0.301)+(0.4771)\\ \approx 1.9821\end{array}$

1. $5X+Y$
2. 1.9821

7. $\log 162$

$\begin{array}{rcl}\log 162& =& \log \left(2·3^4\right)\\ & =& \log 2+\log 3^4\end{array}$

1. $\begin{array}{c}=\log 2+4\log 3\\ =\left(X\right)+4\left(Y\right)\\ =X+4Y\end{array}$
2. $\begin{array}{c}\approx (0.301)+4(0.4771)\\ \approx 2.2094\end{array}$

1. $X+4Y$
2. 2.2094

8. $\log 144$

$\begin{array}{rcl}\log 144& =& \log \left(2^4·3^2\right)\\ & =& \log 2^4+\log 3^2\end{array}$

1. $\begin{array}{c}=4\log 2+2\log 3\\ =4\left(X\right)+2\left(Y\right)\\ =4X+2Y\end{array}$
2. $\begin{array}{c}\approx 4(0.301)+2(0.4771)\\ \approx 2.1582\end{array}$

1. $4X+2Y$
2. 2.1582

9. $6^x=30$

$\begin{array}{rcl}\log 6^x& =& \log 30\\ x \log 6& =& \log 30\end{array}$

$x=\frac{\log 30}{\log 6}\approx 1.8982$

10. ${81}^{x–3}=15$

$\begin{array}{rcl}\log {81}^{x–3}& =& \log 15\\ \left(x–3\right) \log 81& =& \log 15\\ x–3& =& \frac{\log 15}{\log 81}\end{array}$

$x=\frac{\log 15}{\log 81}+3\approx 3.6162$

11. ${25}^{3x+1}=520$

$\begin{array}{rcl}\log 25{ }^{3x+1}& =& \log 520\\ \left(3x+1\right) \log 25& =& \log 520\\ 3x+1& =& \frac{\log 520}{\log 25}\\ 3x& =& \frac{\log 520}{\log 25}–1\end{array}$

$x=\frac{\log 520}{3\log 25}–\frac{1}{3}\approx 0.3143$

12. $7^{2x}=490$

$\begin{array}{rcl}\log 7^{ 2x}& =& \log 490\\ 2x \log 7& =& \log 490\\ 2x& =& \frac{\log 490}{\log 7}\end{array}$

$x=\frac{\log 490}{2\log 7}\approx 1.5916$

13. $2^x=56$

$\begin{array}{rcl}\log 2^x& =& \log 56\\ x \log 2& =& \log 56\end{array}$

$x=\frac{\log 56}{\log 2}\approx 5.8074$

14. ${32}^{x+1}=12$

$\begin{array}{rcl}\log {32}^{x+1}& =& \log 12\\ \left(x+1\right) \log 32& =& \log 12\\ x+1& =& \frac{\log 12}{\log 32}\end{array}$

$x=\frac{\log 12}{\log 32}–1\approx –0.2830$

15. ${\log }_5{9}^4$

$\frac{\log {\left(9\right)}^4}{\log 5}$

$\frac{4\log 9}{\log 5}$

16. ${\log }_{3}11x$

$\frac{\log 11x}{\log 3}$

$\frac{\log 11+\log x}{\log 3}$

17. ${\log }_{2}6+{\log }_{3}4$

$\frac{\log 6}{\log 2}+\frac{\log 4}{\log 3}$

18. $7^{\frac{x+2}{3}}=84$

$\begin{array}{rcl}\log 7^{\frac{x+2}{3}}& =& \log 84\\ \left(\frac{x+2}{3}\right) \log 7& =& \log 84\\ \left(3\right)\frac{x+2}{3}& =& \frac{\log 84}{\log 7}\left(3\right)\\ x+2& =& \frac{3 \log 84}{\log 7}\\ & & \end{array}$

$x=\frac{3\log 84}{\log 7}–2$

19. $2^{4–x}=13$

$\begin{array}{rcl}\log 2^{4–x}& =& \log 13\\ \left(4–x\right) \log 2& =& \log 13\\ 4–x& =& \frac{\log 13}{\log 2}\\ \left(–1\right)\left(–x\right)& =& \left(\frac{\log 13}{\log 2}–4\right)\left(–1\right)\end{array}$

$x=–\frac{\log 13}{\log 2}+4$

20. ${15}^{\frac{x}{7}}=25$

$\begin{array}{rcl}\log {15}^{\frac{x}{7}}& =& \log 25\\ \frac{x}{7} \log 15& =& \log 25\\ \left(7\right)\frac{x}{7}& =& \frac{\log 25}{\log 15}\left(7\right)\end{array}$

$x=\frac{7 \log 25}{\log 15}$