Section 9.11 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.
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.
\(\log(x)+\log(x-2)=\log(15)\)
SolutionThe only solution is 5.
20.
\(\log_2(x+1)-\log_2(x-3)=1\)
SolutionThe only solution is 7.
21.
\(2\log_3(x)-2=\log_3(x-2)\)
SolutionThe solutions are 3 and 6.
22.
\(3^{x+5}-9^{x+2}=0\)
SolutionThe only solution is 1.
23.
\(5^x=3^{2x-1}\)
SolutionThe only solution is \(\frac{\log(3)}{\log\left(\frac{5}{9}\right)}\) which is approximately 1.87.
24.
\(\ln(x+2)-\ln(x-3)=2\)
SolutionThe only solution is \(\frac{3e^2+3}{e^2-1}\) which is approximately 3.78.
25.
\(e^{3x-1}=5\)
SolutionThe only solution is \(\frac{\ln(5)+1}{3}\) which is approximately 0.87.