Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$$\frac { 1 } { 5 } \left[ 5 \ln ( x + 1 ) - \ln x - \ln \left( x ^ { 2 } - 1 \right) \right]$$
A) $\ln \sqrt [ 5 ] { \frac { ( x + 1 ) ^ { 5 } } { x \left( x ^ { 2 } - 1 \right) } }$
B) $\ln \sqrt [ 5 ] { \frac { ( x + 1 ) ^ { 5 } \left( x ^ { 2 } - 1 \right) } { x } }$
C) $\ln \sqrt [ 5 ] { \frac { 5 ( x + 1 ) } { x \left( x ^ { 2 } - 1 \right) } }$
D) $\ln \sqrt [ 5 ] { \frac { x ( x + 1 ) ^ { 5 } } { \left( x ^ { 2 } - 1 \right) } }$

Question

Quizplus · Accepted Answer

Using properties of logarithms, the given expression can be simplified as follows: $\frac{1}{5} [5\ln(x+1) - \ln x - \ln(x^2 - 1)]$ can be rewritten using the property $\ln a - \ln b = \ln(\frac{a}{b})$ and the distributive property of multiplication over addition/subtraction inside the logarithm. This results in $\ln \sqrt[5]{\frac{(x+1)^5}{x(x^2-1)}}$, which matches choice A.

Use Properties of Logarithms to Condense the Logarithmic Expression 15[5ln⁡(x+1)−ln⁡x−ln⁡(x2−1)]\frac { 1 } { 5 } \left[ 5 \ln ( x + 1 ) - \ln x - \ln \left( x ^ { 2 } - 1 \right) \right]51​[5ln(x+1)−lnx−ln(x2−1)]

Use Properties of Logarithms to Condense the Logarithmic Expression $\frac { 1 } { 5 } \left[ 5 \ln ( x + 1 ) - \ln x - \ln \left( x ^ { 2 } - 1 \right) \right]$