Algebra Examples

$\log_{3} (\frac{1}{27}) = x$

Step 1

Rewrite the equation as $x = \log_{3} (\frac{1}{27})$ .

$x = \log_{3} (\frac{1}{27})$

Step 2

Logarithm base $3$ of $\frac{1}{27}$ is $- 3$ .

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Step 2.1

Rewrite as an equation.

$x = \log_{3} (\frac{1}{27}) = x$

Step 2.2

Rewrite $\log_{3} (\frac{1}{27}) = x$ 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 $b^{y} = x$ .

$x = 3^{x} = \frac{1}{27}$

Step 2.3

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

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

Step 2.4

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

$x = x = - 3$

Step 2.5

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

$x = - 3$