Evaluate log base 3 of log base 4 of 64

$\log_{3} (\log_{4} (64))$

Logarithm base $4$ of $64$ is $3$ .

Rewrite as an equation.

$\log_{3} (\log_{4} (64) = x)$

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

$\log_{3} (4^{x} = 64)$

Create expressions in the equation that all have equal bases.

$\log_{3} ({(2^{2})}^{x} = 2^{6})$

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

$\log_{3} (2^{2 x} = 2^{6})$

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

$\log_{3} (2 x = 6)$

Solve for $x$ .

$\log_{3} (x = 3)$

The variable $x$ is equal to $3$ .

$\log_{3} (3)$

Logarithm base $3$ of $3$ is $1$ .

$1$

Evaluate log base 3 of log base 4 of 64

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