Precalculus Examples

$\log_{5} (\frac{\sqrt{5}}{25}) + e^{\ln (3) + \ln (4)}$

Step 1

Simplify each term.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{5} (\frac{\sqrt{5}}{25}) + e^{\ln (3 \cdot 4)}$

Step 1.2

Exponentiation and log are inverse functions.

$\log_{5} (\frac{\sqrt{5}}{25}) + 3 \cdot 4$

Step 1.3

Multiply $3$ by $4$ .

$\log_{5} (\frac{\sqrt{5}}{25}) + 12$

Step 2

Rewrite $\log_{5} (\frac{\sqrt{5}}{25})$ using the change of base formula.

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Step 2.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + 12$

Step 2.2

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

$\frac{\log (\frac{\sqrt{5}}{25})}{\log (5)} + 12$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{\sqrt{5}}{25})}{\log (5)} + 12$

Decimal Form:

$10.5$