Linear Algebra Examples

$\log_{36} (\frac{1}{6}) = x$

Step 1

Rewrite the equation as $x = \log_{36} (\frac{1}{6})$ .

$x = \log_{36} (\frac{1}{6})$

Step 2

Logarithm base $36$ of $\frac{1}{6}$ is $- \frac{1}{2}$ .

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Step 2.1

Rewrite as an equation.

$x = \log_{36} (\frac{1}{6}) = x$

Step 2.2

Rewrite $\log_{36} (\frac{1}{6}) = x$ 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 $b^{y} = x$ .

$x = 36^{x} = \frac{1}{6}$

Step 2.3

Create expressions in the equation that all have equal bases.

$x = {(6^{2})}^{x} = 6^{- 1}$

Step 2.4

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

$x = 6^{2 x} = 6^{- 1}$

Step 2.5

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

$x = 2 x = - 1$

Step 2.6

Solve for $x$ .

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

Step 2.7

The variable $x$ is equal to $- \frac{1}{2}$ .

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

Step 3

The domain is the set of all valid $x$ values.

${x | x = - \frac{1}{2}}$

Step 4