Precalculus Examples

$\log_{8} (16^{\log (\frac{1}{10000})})$

Step 1

Logarithm base $10$ of $\frac{1}{10000}$ is $- 4$ .

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Step 1.1

Rewrite as an equation.

$\log_{8} (16^{\log (\frac{1}{10000}) = x})$

Step 1.2

Rewrite $\log (\frac{1}{10000}) = x$ 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 $b^{y} = x$ .

$\log_{8} (16^{10^{x} = \frac{1}{10000}})$

Step 1.3

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

$\log_{8} (16^{10^{x} = 10^{- 4}})$

Step 1.4

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

$\log_{8} (16^{x = - 4})$

Step 1.5

The variable $x$ is equal to $- 4$ .

$\log_{8} (16^{- 4})$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log_{8} (\frac{1}{16^{4}})$

Step 3

Raise $16$ to the power of $4$ .

$\log_{8} (\frac{1}{65536})$

Step 4

Logarithm base $8$ of $\frac{1}{65536}$ is $- \frac{16}{3}$ .

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Step 4.1

Rewrite as an equation.

$\log_{8} (\frac{1}{65536}) = x$

Step 4.2

Rewrite $\log_{8} (\frac{1}{65536}) = x$ 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 $b^{y} = x$ .

$8^{x} = \frac{1}{65536}$

Step 4.3

Create expressions in the equation that all have equal bases.

${(2^{3})}^{x} = 2^{- 16}$

Step 4.4

Rewrite ${(2^{3})}^{x}$ as $2^{3 x}$ .

$2^{3 x} = 2^{- 16}$

Step 4.5

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

$3 x = - 16$

Step 4.6

Solve for $x$ .

$x = - \frac{16}{3}$

Step 4.7

The variable $x$ is equal to $- \frac{16}{3}$ .

$- \frac{16}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{16}{3}$

Decimal Form:

$- 5.\overline{3}$

Mixed Number Form:

$- 5 \frac{1}{3}$