Precalculus Examples

$\log_{81} (3) = x$

Step 1

Rewrite the equation as $x = \log_{81} (3)$ .

$x = \log_{81} (3)$

Step 2

Logarithm base $81$ of $3$ is $\frac{1}{4}$ .

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Step 2.1

Rewrite as an equation.

$x = \log_{81} (3) = x$

Step 2.2

Rewrite $\log_{81} (3) = x$ 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 $b^{y} = x$ .

$x = 81^{x} = 3$

Step 2.3

Create expressions in the equation that all have equal bases.

$x = {(3^{4})}^{x} = 3^{1}$

Step 2.4

Rewrite ${(3^{4})}^{x}$ as $3^{4 x}$ .

$x = 3^{4 x} = 3^{1}$

Step 2.5

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

$x = 4 x = 1$

Step 2.6

Solve for $x$ .

$x = x = \frac{1}{4}$

Step 2.7

The variable $x$ is equal to $\frac{1}{4}$ .

$x = \frac{1}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$