Precalculus Examples

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

Step 1

Logarithm base $64$ of $4$ is $\frac{1}{3}$ .

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Step 1.1

Rewrite as an equation.

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

Step 1.2

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

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

Step 1.3

Create expressions in the equation that all have equal bases.

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

Step 1.4

Rewrite ${(2^{6})}^{x}$ as $2^{6 x}$ .

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

Step 1.5

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

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

Step 1.6

Solve for $x$ .

$\log_{3} (x = \frac{1}{3})$

Step 1.7

The variable $x$ is equal to $\frac{1}{3}$ .

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Step 1.7.1

Factor $2$ out of $2$ .

$\log_{3} (\frac{2 (1)}{6})$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $2$ out of $6$ .

$\log_{3} (\frac{2 \cdot 1}{2 \cdot 3})$

Step 1.7.2.2

Cancel the common factor.

$\log_{3} (\frac{2 \cdot 1}{2 \cdot 3})$

Step 1.7.2.3

Rewrite the expression.

$\log_{3} (\frac{1}{3})$

Step 2

Logarithm base $3$ of $\frac{1}{3}$ is $- 1$ .

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Step 2.1

Rewrite as an equation.

$\log_{3} (\frac{1}{3}) = x$

Step 2.2

Rewrite $\log_{3} (\frac{1}{3}) = x$ 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 $b^{y} = x$ .

$3^{x} = \frac{1}{3}$

Step 2.3

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

$3^{x} = 3^{- 1}$

Step 2.4

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

$x = - 1$

Step 2.5

The variable $x$ is equal to $- 1$ .

$- 1$