Properties of Logs Practice

Evaluate the logarithm using the change-of-0.base formula. (Round your answer to three decimal places):  $\log_74$  

Answer explanation

 $\log_bx=\frac{\log x}{\log b}=\frac{\ln x}{\ln b}$  

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Change-of-base

Evaluate the logarithm using the change-of-0.base formula. (Round your answer to three decimal places):  $\log_{20}0.125$  

Answer explanation

 $\log_bx=\frac{\log x}{\log b}=\frac{\ln x}{\ln b}$  

Rewrite the log as a ratio of common logarithms: $\log_3x$  

•  $\frac{\log3}{\log x}$  

•  $\frac{\log x}{\log3}$  

Answer explanation

Think: Change-of-base

Rewrite the log as a ratio of natural logarithms: $\log_x\ \frac{3}{4}$  

•  $\frac{\ln\ \frac{3}{4}}{\ln\ x}$  

•  $\frac{\ln x}{\ln\ \frac{3}{4}}$  

Answer explanation

Change-of-base

Use the properties of logarithms to write this expression as a sum, difference, and/or constant multiple of logarithms (aka EXPAND the expression): $\log\ \frac{y}{2}$  

•  $\log y\cdot\log2$  

•  $\frac{\log y}{\log2}$  

•  $\log y-\log2$  

•  $\log y^2$  

Answer explanation

Use the properties of logarithms to write this expression as a sum, difference, and/or constant multiple of logarithms (aka EXPAND the expression):  $\ln^3\sqrt{t}\ \ \ \left(cube\ root\right)$  

•  $\ln t+\ln\ \frac{1}{3}$  

•  $3\ln t$  

•  $\ln t^{\frac{1}{3}}$  

•  $\frac{1}{3}\ln t$  

Answer explanation

Rewrite the cube root as t^1/3 first.  Then, use the power property of logs.

Use the properties of logarithms to write this expression as a sum, difference, and/or constant multiple of logarithms (aka EXPAND the expression):  $\ln\ \frac{x}{\sqrt{x^2+1}}$  

•  $\ln x+\ln\sqrt{x^2+1}$  

•  $\ln x-\frac{1}{2}\ln\left(x^2+1\right)$  

•  $\ln x-\ln\sqrt{x^2+1}$  

•  $\frac{\ln x}{\frac{1}{2}\ln\left(x^2+1\right)}$  

Answer explanation

The square root is the 1/2 power.

Expand the expression using the properties of logarithms.

•  $\frac{1}{3}\ln x-\frac{1}{3}\ln y$  

•  $3\ln x-3\ln y$  

•  $\frac{1}{3}\ln\left(\frac{x}{y}\right)$  

•  $\frac{1}{3}\ln x-\ln y$  

Answer explanation

Power prop first, then quotient prop.

Expand the expression using the properties of logarithms.

• $2\ln x-3\ln y$

• $6\ln xy$

• $\ln x-\frac{3}{2}\ln y$

• $6\ln x-6\ln y$

Answer explanation

$Use\ the\ power\ property\ twice,\ and\ quotient\ property$

Expand the expression:

• $\ln x+\frac{1}{2}\ln\left(x+2\right)$

• $\ln x^2\cdot\frac{1}{2}+\ln\ \frac{1}{2}\left(x+2\right)$

• $2\ln x+\ln x+\ln2$

• $\frac{1}{2}\ln x^2+\frac{1}{2}\ln\left(x+2\right)$