Evaluate log base 2 of log base 4 of 16

$\log_{2} (\log_{4} (16))$

Step 1

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

Rewrite as an equation.

$\log_{2} (\log_{4} (16) = x)$

Rewrite $\log_{4} (16) = x$ 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 $b^{y} = x$ .

$\log_{2} (4^{x} = 16)$

Create expressions in the equation that all have equal bases.

$\log_{2} ({(2^{2})}^{x} = 2^{4})$

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

$\log_{2} (2^{2 x} = 2^{4})$

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

$\log_{2} (2 x = 4)$

Solve for $x$ .

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

The variable $x$ is equal to $2$ .

$\log_{2} (2)$

Step 2

Logarithm base $2$ of $2$ is $1$ .

$1$

Evaluate log base 2 of log base 4 of 16

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