4.5E: Exercises

4.5: Logarithmic Properties

Verbal

1) How does the power rule for logarithms help when solving logarithms with the form \(\log _b(\sqrt[n]{x})\)$?$

Answer

Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, \(\log _b \left ( x^{\frac{1}{n}} \right ) = \dfrac{1}{n}\log_{b}(x)\).

2) What does the change-of-base formula do? Why is it useful when using a calculator?

Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

3) \(\log _b (7x\cdot 2y)\)

Answer

\(\log _b (2)+\log _b (7)+\log _b (x)+\log _b (y)\)

4) \(\ln (3ab\cdot 5c)\)

5) \(\log_b \left ( \dfrac{13}{17} \right )\)

Answer

\(\log _b (13)-\log _b (17)\)

6) \(\log_4 \left ( \dfrac{\frac{x}{z}}{w} \right )\)

7) \(\ln \left ( \dfrac{1}{4^k} \right )\)

Answer

\(-k\ln(4)\)

8) \(\log _2 (y^x)\)

For the following exercises, condense to a single logarithm if possible.

9) \(\ln (7)+\ln (x)+\ln (y)\)

Answer

\(\ln(7xy)\)

10) \(\log_3(2)+\log_3(a)+\log_3(11)+\log_3(b)\)

11) \(\log_b(28)-\log_b(7)\)

Answer

\(\log_b(4)\)

12) \(\ln (a)-\ln (d)-\ln (c)\)

13) \(-\log_b\left ( \dfrac{1}{7} \right )\)

Answer

\(\log_b(7)\)

14) \(\dfrac{1}{3}\ln(8)\)

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

15) \(\log \left ( \dfrac{x^{15}y^{13}}{z^{19}} \right )\)

Answer

\(15\log (x)+13\log (y)-19\log (z)\)

16) \(\ln \left ( \frac{a^{-2}}{b^{-4}c^{5}} \right )\)

17) \(\log \left ( \sqrt{x^3y^{-4}} \right )\)

Answer

\(\frac{3}{2}\log (x)-2\log (y)\)

18) \(\ln \left ( y\sqrt{\frac{y}{1-y}} \right )\)

19) \(\log \left ( x^2y^3 \sqrt[3]{x^2y^5} \right )\)

Answer

\(\dfrac{8}{3}\log (x)+\dfrac{14}{3}\log (y)\)

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

20) \(\log \left ( 2x^4 \right )+\log \left (3x^5 \right )\)

21) \(\ln \left ( 6x^9 \right )-\ln \left (3x^2 \right )\)

Answer

\(\ln \left ( 2x^7 \right )\)

22) \(2\log (x)+3\log (x+1)\)

23) \(\log (x)-\dfrac{1}{2}\log (y)+3\log (z)\)

Answer

\(\log \left ( \dfrac{xz^3}{\sqrt{y}} \right )\)

24) \(4\log _7(c)+\dfrac{\log _7(a)}{3}+\dfrac{\log _7(b)}{3}\)

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

25) \(\log _7(15)\) to base \(e\)

Answer

\(\log _7(15)=\dfrac{\ln (15)}{\ln (7)}\)

26) \(\log _{14}(55.875)\) to base \(10\)

For the following exercises, suppose \(\log _5(6)=a\) and \(\log _5(11)=b\) $<strong>.</strong>\text{<strong>\hspace{0.17em}</strong>}$Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b\) $<strong>.</strong>\text{<strong>\hspace{0.17em}</strong>}$Show the steps for solving.

27) \(\log _{11} (5)\)

Answer

\(\log _{11} (5)=\dfrac{\log_5 (5)}{\log_5 (11)}=\dfrac{1}{b}\)

28) \(\log _{6} (55)\)

29) \(\log _{11}\left (\dfrac{6}{11} \right )\)

Answer

\(\log _{11}\left (\dfrac{6}{11} \right )=\dfrac{\log _{11}\left (\frac{6}{11} \right )}{\log _{5}(11)}=\dfrac{\log _{5}(6)-\log _{5}(11)}{\log _{5}(11)}=\dfrac{a-b}{b}=\dfrac{a}{b}-1\)

Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

30) \(\log _3 \left ( \dfrac{1}{9} \right )-3\log _3 (3)\)

31) \(6\log _8 (2)+\dfrac{\log _8 (64)}{3\log _8 (4)}\)

Answer

\(3\)

32) \(2\log _9 (3)-4\log _9 (3)+\log _9 \left (\dfrac{1}{729} \right )\)

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

33) \(\log _3 (22)\)

Answer

\(2.81359\)

34) \(\log _8 (65)\)

35) \(\log _6 (5.38)\)

Answer

\(0.93913\)

36) \(\log _4 \left (\dfrac{15}{2} \right )\)

37) \(\log _{\frac{1}{2}} (4.7)\)

Answer

\(-2.23266\)

Extensions

38) Use the product rule for logarithms to find all \(x\) values such that \(\log _{12} (2x+6)+\log _{12} (x+2)=2\)$\mathrm{.}$Show the steps for solving.

39) Use the quotient rule for logarithms to find all \(x\) values such that \(\log _{6} (x+2)-\log _{6} (x-3)=1\)$\mathrm{.}$Show the steps for solving.

Answer

Rewriting as an exponential equation and solving for \(x\):

\(\begin{align*}
6^1 &= \frac{x+2}{x-3}\\
0 &= \frac{x+2}{x-3}-6\\
0 &= \frac{x+2}{x-3}-\frac{6(x-3)}{(x-3)}\\
0 &= \frac{x+2-6x+18}{x-3}\\
0 &= \frac{x-4}{x-3}\\
x &= 4
\end{align*}\)

Checking, we find that \(\log _6(4+2)-\log _6(4-3)=\log _6(6)-\log _6(1)\) is defined, so \(x=4\)

40) Can the power property of logarithms be derived from the power property of exponents using the equation \(b^x=m\)$?$If not, explain why. If so, show the derivation.

41) Prove that \(\log_b(n)=\frac{1}{\log_b(n)}\) for any positive integers \(b>1\) and \(n>1\)$\mathrm{.}$

Answer

Let \(b\) and \(n\) be positive integers greater than \(1\)$\mathrm{.}$Then, by the change-of-base formula, \(\log_b(n)=\frac{\log_n(n)}{\log_n(b)}=\frac{1}{\log_n(b)}\)

42) Does \(\log_{81}(2401)=\log_3(7)\)$?$Verify the claim algebraically.