Precalculus Examples

$\log_{2} (\log_{2} (\log_{2} (2^{4 x}))) = 3$

Step 1

Expand $\log_{2} (\log_{2} (\log_{2} (2^{4 x})))$ .

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Step 1.1

Expand $\log_{2} (2^{4 x})$ by moving $4 x$ outside the logarithm.

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

Step 1.2

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

$\log_{2} (\log_{2} (4 x \cdot 1))$

Step 1.3

Multiply $4$ by $1$ .

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

Step 2

The expanded equation is $\log_{2} (\log_{2} (4 x)) = 3$ .

$\log_{2} (\log_{2} (4 x)) = 3$

Step 3

Rewrite $\log_{2} (\log_{2} (4 x)) = 3$ 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 $b^{y} = x$ .

$2^{3} = \log_{2} (4 x)$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\log_{2} (4 x) = 2^{3}$ .

$\log_{2} (4 x) = 2^{3}$

Step 4.2

Rewrite $\log_{2} (4 x) = 2^{3}$ 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 $b^{y} = x$ .

$2^{2^{3}} = 4 x$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Rewrite the equation as $4 x = 2^{2^{3}}$ .

$4 x = 2^{2^{3}}$

Step 4.3.2

Divide each term in $4 x = 2^{2^{3}}$ by $4$ and simplify.

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Step 4.3.2.1

Divide each term in $4 x = 2^{2^{3}}$ by $4$ .

$\frac{4 x}{4} = \frac{2^{2^{3}}}{4}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $4$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2^{2^{3}}}{4}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2^{2^{3}}}{4}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Simplify the numerator.

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Step 4.3.2.3.1.1

Raise $2$ to the power of $3$ .

$x = \frac{2^{8}}{4}$

Step 4.3.2.3.1.2

Raise $2$ to the power of $8$ .

$x = \frac{256}{4}$

Step 4.3.2.3.2

Divide $256$ by $4$ .

$x = 64$