Algebra Examples

\log_{81} (3)

$\log_{81} (3)$

Step 1

Rewrite as an equation.

\log_{81} (3) = x

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

Step 2

Rewrite

\log_{81} (3) = x

$\log_{81} (3) = x$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$ does not equal

1

$1$ , then

\log_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

81^{x} = 3

$81^{x} = 3$

Step 3

Create expressions in the equation that all have equal bases.

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

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

Step 4

Rewrite

{(3^{4})}^{x}

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

3^{4 x}

$3^{4 x}$ .

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

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

Step 5

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

4 x = 1

$4 x = 1$

Step 6

Solve for

x

$x$ .

x = \frac{1}{4}

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

Step 7

The variable

x

$x$ is equal to

\frac{1}{4}

$\frac{1}{4}$ .

\frac{1}{4}

$\frac{1}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

\frac{1}{4}

$\frac{1}{4}$

Decimal Form:

0.25

$0.25$