Precalculus Examples

$\log_{5} (5) - \log_{2} (\sqrt{2}) + \ln (e^{3})$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $5$ of $5$ is $1$ .

$1 - \log_{2} (\sqrt{2}) + \ln (e^{3})$

Step 1.2

Use logarithm rules to move $3$ out of the exponent.

$1 - \log_{2} (\sqrt{2}) + 3 \ln (e)$

Step 1.3

The natural logarithm of $e$ is $1$ .

$1 - \log_{2} (\sqrt{2}) + 3 \cdot 1$

Step 1.4

Multiply $3$ by $1$ .

$1 - \log_{2} (\sqrt{2}) + 3$

Step 2

Add $1$ and $3$ .

$4 - \log_{2} (\sqrt{2})$

Step 3

Rewrite $\log_{2} (\sqrt{2})$ using the change of base formula.

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Step 3.1

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 $0$ .

$4 - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$4 - \frac{\log (\sqrt{2})}{\log (2)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$4 - \frac{\log (\sqrt{2})}{\log (2)}$

Decimal Form:

$3.5$