## Section8.11Additional Practice Related to Logarithms and Exponential Equations

### ExercisesExercises

Determine the value of each logarithm without the use of a calculator.

###### 1.

$\log_5\left(\frac{1}{5}\right)$

Solution

$\log_5\left(\frac{1}{5}\right)=-1$

###### 2.

$\log_3(81)$

Solution

$\log_3(81)=4$

###### 3.

$\log_{100}(10)$

Solution

$\log_{100}(10)=\frac{1}{2}$

###### 4.

$\log_{27}(9)$

Solution

$\log_{27}(9)=\frac{2}{3}$

###### 5.

$\log_4\left(\frac{1}{2}\right)$

Solution

$\log_4\left(\frac{1}{2}\right)=-\frac{1}{2}$

###### 6.

$\log(10^{-6})$

Solution

$\log(10^{-6})=-6$

Use the change of base formula and a calculator to estimate the value of each logarithm. Round each value to three digits after the decimal point.

###### 7.

$\log_7(8)$

Solution

$\log_7(8) \approx 1.069$

###### 8.

$\log_{12}(100)$

Solution

$\log_{12}(100) \approx 1.852$

###### 9.

$\log_{19}\left(\frac{2}{87}\right)$

Solution

$\log_{19}\left(\frac{2}{87}\right) \approx -1.281$

###### 10.

$\log_2(1977)$

Solution

$\log_2(1977) \approx 10.949$

Completely expand each logarithmic expression.

###### 11.

$\log\left(\frac{xy^2}{4\sqrt{z}}\right)$

Solution

$\log(x)+2\log(y)-\log(4)-\frac{1}{2}\log(z)$

###### 12.

$\log_2\left(\frac{8\sqrt[3]{x^7}}{y^9z^4}\right)$

Solution

$3+\frac{7}{3}\log_2(x)-9\log_2(y)-4\log_2(z)$

###### 13.

$\log_8\left((x+y)(x-y)\right)$

Solution

$\log_8(x+y)+\log_8(x-y)$

###### 14.

$\log\left(\frac{x^2+y^2}{x+y}\right)$

Solution

$\log\left(x^2+y^2\right)-\log(x+y)$

Combine each expression into a single logarithmic expression.

###### 15.

$\log_2(x)-4\log_2(y)-7\log_2(z)$

Solution

$\log_2\left(\frac{x}{y^4z^7}\right)$

###### 16.

$3+\frac{1}{5}\log(x)-10\log(y)+14\log(z)$

Solution

$\log\left(\frac{1000z^{14}\sqrt[5]{x}}{y^{10}}\right)$

###### 17.

$-\frac{1}{2}-\frac{2}{3}\log_9(x)+\log_9(y-z)$

Solution

$\log_9\left(\frac{y-z}{3\sqrt[3]{x^2}}\right)$

###### 18.

$8\log_6\left(\sqrt{x}\right)-2-4\log_6(x)-\frac{4}{9}\log_6(y)$

Solution

$\log_6\left(\frac{1}{36\sqrt[9]{y^4}}\right)$

Determine all solutions to each stated equation. State the exact solutions and , where appropriate, also state approximate solutions (rounded to the nearest hundredth).

###### 19.

$\log(x)+\log(x-2)=\log(15)$

Solution

The only solution is 5.

###### 20.

$\log_2(x+1)-\log_2(x-3)=1$

Solution

The only solution is 7.

###### 21.

$2\log_3(x)-2=\log_3(x-2)$

Solution

The solutions are 3 and 6.

###### 22.

$3^{x+5}-9^{x+2}=0$

Solution

The only solution is 1.

###### 23.

$5^x=3^{2x-1}$

Solution

The only solution is $\frac{\log(3)}{\log\left(\frac{5}{9}\right)}$ which is approximately 1.87.

###### 24.

$\ln(x+2)-\ln(x-3)=2$

Solution

The only solution is $\frac{3e^2+3}{e^2-1}$ which is approximately 3.78.

###### 25.

$e^{3x-1}=5$

Solution

The only solution is $\frac{\ln(5)+1}{3}$ which is approximately 0.87.