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Section 8.11 Additional Practice Related to Logarithms and Exponential Equations

Exercises Exercises

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.