Precalculus Examples

$\log_{2} (16 e^{\ln (8)})$

Step 1

Rewrite $\log_{2} (16 e^{\ln (8)})$ as $\log_{2} (16) + \log_{2} (e^{\ln (8)})$ .

$\log_{2} (16) + \log_{2} (e^{\ln (8)})$

Step 2

Simplify each term.

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Step 2.1

Logarithm base $2$ of $16$ is $4$ .

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Step 2.1.1

Rewrite as an equation.

$\log_{2} (16) = x + \log_{2} (e^{\ln (8)})$

Step 2.1.2

Rewrite $\log_{2} (16) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{x} = 16 + \log_{2} (e^{\ln (8)})$

Step 2.1.3

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{4} + \log_{2} (e^{\ln (8)})$

Step 2.1.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = 4 + \log_{2} (e^{\ln (8)})$

Step 2.1.5

The variable $x$ is equal to $4$ .

$4 + \log_{2} (e^{\ln (8)})$

Step 2.2

Expand $\log_{2} (e^{\ln (8)})$ by moving $\ln (8)$ outside the logarithm.

$4 + \ln (8) \log_{2} (e)$

Step 2.3

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$4 + \ln (2^{3}) \log_{2} (e)$

Step 2.4

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$4 + 3 \ln (2) \log_{2} (e)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$4 + 3 \ln (2) \log_{2} (e)$

Decimal Form:

$7$