Evaluate log base 64 of 16

$\log_{64} (16)$

Rewrite as an equation.

$\log_{64} (16) = x$

Rewrite $\log_{64} (16) = x$ 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 $b^{y} = x$ .

$64^{x} = 16$

Create expressions in the equation that all have equal bases.

${(2^{6})}^{x} = 2^{4}$

Rewrite ${(2^{6})}^{x}$ as $2^{6 x}$ .

$2^{6 x} = 2^{4}$

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

$6 x = 4$

Solve for $x$ .

$x = \frac{2}{3}$

The variable $x$ is equal to $\frac{2}{3}$ .

Factor $2$ out of $4$ .

$\frac{2 (2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 \cdot 2}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 3}$

Rewrite the expression.

$\frac{2}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$

Evaluate log base 64 of 16

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