Precalculus Examples

$\log_{16} ({(\frac{1}{4})}^{3})$

Step 1

Apply the product rule to $\frac{1}{4}$ .

$\log_{16} (\frac{1^{3}}{4^{3}})$

Step 2

One to any power is one.

$\log_{16} (\frac{1}{4^{3}})$

Step 3

Raise $4$ to the power of $3$ .

$\log_{16} (\frac{1}{64})$

Step 4

Logarithm base $16$ of $\frac{1}{64}$ is $- \frac{3}{2}$ .

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Step 4.1

Rewrite as an equation.

$\log_{16} (\frac{1}{64}) = x$

Step 4.2

Rewrite $\log_{16} (\frac{1}{64}) = x$ 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 $b^{y} = x$ .

$16^{x} = \frac{1}{64}$

Step 4.3

Create expressions in the equation that all have equal bases.

${(2^{4})}^{x} = 2^{- 6}$

Step 4.4

Rewrite ${(2^{4})}^{x}$ as $2^{4 x}$ .

$2^{4 x} = 2^{- 6}$

Step 4.5

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

$4 x = - 6$

Step 4.6

Solve for $x$ .

$x = - \frac{3}{2}$

Step 4.7

The variable $x$ is equal to $- \frac{3}{2}$ .

$- \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$

Mixed Number Form:

$- 1 \frac{1}{2}$