Evaluate log base 64 of 4

$\log_{64} (4)$

Step 1

Rewrite as an equation.

$\log_{64} (4) = x$

Step 2

Rewrite $\log_{64} (4) = x$ 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 $b^{y} = x$ .

$64^{x} = 4$

Step 3

Create expressions in the equation that all have equal bases.

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

Step 4

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

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

Step 5

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

$6 x = 2$

Step 6

Solve for $x$ .

$x = \frac{1}{3}$

Step 7

The variable $x$ is equal to $\frac{1}{3}$ .

Factor $2$ out of $2$ .

$\frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3}$

Rewrite the expression.

$\frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$

Evaluate log base 64 of 4

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