Questions: Write the expression as a sum and/or difference of logarithms. Express powers as factors. log5(25 x) log5(25 x) = (Type an exact answer in simplified form.)

Write the expression as a sum and/or difference of logarithms. Express powers as factors.
log5(25 x)
log5(25 x) = (Type an exact answer in simplified form.)

Transcript text: Write the expression as a sum and/or difference of logarithms. Express powers as factors. \[ \log _{5}(25 x) \] $\log _{5}(25 x)=$ $\square$ (Type an exact answer in simplified form.)

Solution

Solution Steps

Step 1: Identify the Coefficient and Base

The given logarithmic expression is $\log_{5}(25x)$. Here, the base of the logarithm is $5$, and the coefficient of the variable inside the logarithm is $25$.

Step 2: Apply Logarithmic Properties

Applying the product rule $\log_b(MN) = \log_b(M) + \log_b(N)$, we can separate the coefficient $a$ and the variable $x$: $\log_{5}(25x) = \log_{5}(25) + \log_{5}(x)$.

Step 3: Evaluate Logarithmic Expressions

If $a$ or $x$ are powers of $b$, they can be directly evaluated. Otherwise, this step might not simplify further without specific values.

Step 4: Simplify

The expression is already in its simplified form as a sum of logarithms, unless specific values allow for further simplification.