Evaluate 5* log base 5 of 4- log base 5 of 16

$5 \cdot \log_{5} (4) - \log_{5} (16)$

Step 1

Simplify each term.

Simplify $5 \log_{5} (4)$ by moving $5$ inside the logarithm.

$\log_{5} (4^{5}) - \log_{5} (16)$

Raise $4$ to the power of $5$ .

$\log_{5} (1024) - \log_{5} (16)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{1024}{16})$

Step 3

Divide $1024$ by $16$ .

$\log_{5} (64)$

Step 4

Rewrite $\log_{5} (64)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (64)}{\log (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (64)}{\log (5)}$

Decimal Form:

$2.58405934 \dots$