Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable
represents a positive real number.
-$\log \left( \frac { 1,000 } { \sqrt { x ^ { 2 } + 25 } } \right)$
A) $3 - \frac { 1 } { 2 } \log \left( x ^ { 2 } + 25 \right)$
B) $3 + \frac { 1 } { 2 } \log \left( x ^ { 2 } + 25 \right)$
C) $3 - \log \left( \sqrt { x ^ { 2 } + 25 } \right)$
D) $3 - \log ( x + 5 )$

Question

Quizplus · Accepted Answer

The given expression can be simplified using the properties of logarithms. The quotient inside the logarithm can be written as the difference of two logarithms: $\log(1000) - \log(\sqrt{x^2 + 25})$. Since $\log(1000) = 3$ (because $10^3 = 1000$) and $\log(\sqrt{x^2 + 25})$ can be written as $\frac{1}{2}\log(x^2 + 25)$ (using the property $\log(a^b) = b\log(a)$), the expression simplifies to $3 - \frac{1}{2}\log(x^2 + 25)$.

Write as the Sum or Difference of Logarithms and Fully log⁡(1,000x2+25)\log \left( \frac { 1,000 } { \sqrt { x ^ { 2 } + 25 } } \right)log(x2+25​1,000​)

Write as the Sum or Difference of Logarithms and Fully $\log \left( \frac { 1,000 } { \sqrt { x ^ { 2 } + 25 } } \right)$