Precalculus Examples

$y = \log_{\frac{1}{4}} (\frac{1}{64})$

Step 1

Rewrite as an equation.

$y = \log_{\frac{1}{4}} (\frac{1}{64}) = x$

Step 2

Rewrite $\log_{\frac{1}{4}} (\frac{1}{64}) = x$ 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 $b^{y} = x$ .

$y = {(\frac{1}{4})}^{x} = \frac{1}{64}$

Step 3

Create expressions in the equation that all have equal bases.

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

Step 4

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

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

Step 5

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

$y = - 2 x = - 6$

Step 6

Solve for $x$ .

$y = x = 3$

Step 7

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

$y = 3$