5.6E: 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.