Precalculus Examples

$\log_{2} (\log_{4} (\log_{2} (x))) = 0$

Step 1

Rewrite $\log_{2} (\log_{4} (\log_{2} (x))) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{0} = \log_{4} (\log_{2} (x))$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $\log_{4} (\log_{2} (x)) = 2^{0}$ .

$\log_{4} (\log_{2} (x)) = 2^{0}$

Step 2.2

Rewrite $\log_{4} (\log_{2} (x)) = 2^{0}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$4^{2^{0}} = \log_{2} (x)$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $\log_{2} (x) = 4^{2^{0}}$ .

$\log_{2} (x) = 4^{2^{0}}$

Step 2.3.2

Rewrite $\log_{2} (x) = 4^{2^{0}}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{4^{2^{0}}} = x$

Step 2.3.3

Solve for $x$ .

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Step 2.3.3.1

Rewrite the equation as $x = 2^{4^{2^{0}}}$ .

$x = 2^{4^{2^{0}}}$

Step 2.3.3.2

Simplify $2^{4^{2^{0}}}$ .

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Step 2.3.3.2.1

Anything raised to $0$ is $1$ .

$x = 2^{4^{1}}$

Step 2.3.3.2.2

Evaluate the exponent.

$x = 2^{4}$

Step 2.3.3.2.3

Raise $2$ to the power of $4$ .

$x = 16$