# 4.5E: Exercises

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## 4.5: Logarithmic Properties

### Verbal

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

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

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

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

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

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

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

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

$$-k\ln(4)$$

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

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

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

$$\ln(7xy)$$

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

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

$$\log_b(4)$$

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

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

$$\log_b(7)$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $$e$$

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

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

For the following exercises, suppose $$\log _5(6)=a$$ and $$\log _5(11)=b$$$.$Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of $$a$$ and $$b$$$.$Show the steps for solving.

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

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

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

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

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

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

$$3$$

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) $$\log _3 (22)$$

$$2.81359$$

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

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

$$0.93913$$

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

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

$$-2.23266$$

### Extensions

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

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

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 $$n>1$$$.$

Let $$b$$ and $$n$$ be positive integers greater than $$1$$$.$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.