Write $5 \ln ( x + 7 ) - 3 \ln x - 4 \ln \left( x ^ { 2 } + 8 \right)$ as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.
A) $\ln \left( x ^ { 3 } ( x + 7 ) ^ { 5 } \left( x ^ { 2 } + 8 \right) ^ { 4 } \right)$
B) $\ln \left( \frac { 5 ( x + 7 ) } { 12 x \left( x ^ { 2 } + 8 \right) } \right)$
C) $\ln \left( 60 x ( x + 7 ) \left( x ^ { 2 } + 8 \right) \right)$
D) $\ln \left( - 4 x ^ { 2 } + 2 x + 3 \right)$
E) $\ln \left( \frac { ( x + 7 ) ^ { 5 } } { x ^ { 3 } \left( x ^ { 2 } + 8 \right) ^ { 4 } } \right)$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be simplified to a single logarithm by combining the terms. The coefficients become exponents, and subtraction in the logarithms translates to division inside the logarithm, leading to the correct answer.

Write 5ln⁡(x+7)−3ln⁡x−4ln⁡(x2+8)5 \ln ( x + 7 ) - 3 \ln x - 4 \ln \left( x ^ { 2 } + 8 \right)5ln(x+7)−3lnx−4ln(x2+8)

Write $5 \ln ( x + 7 ) - 3 \ln x - 4 \ln \left( x ^ { 2 } + 8 \right)$