Precalculus Examples

$\log_{3} (\log_{2} (8))$

Step 1

Logarithm base $2$ of $8$ is $3$ .

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Step 1.1

Rewrite as an equation.

$\log_{3} (\log_{2} (8) = x)$

Step 1.2

Rewrite $\log_{2} (8) = x$ 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 $b^{y} = x$ .

$\log_{3} (2^{x} = 8)$

Step 1.3

Create equivalent expressions in the equation that all have equal bases.

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

Step 1.4

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

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

Step 1.5

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

$\log_{3} (3)$

Step 2

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

$1$