Evaluate log base 81 of 9

$\log_{81} (9)$

Step 1

Rewrite as an equation.

$\log_{81} (9) = x$

Step 2

Rewrite $\log_{81} (9) = x$ 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 $b^{y} = x$ .

$81^{x} = 9$

Step 3

Create expressions in the equation that all have equal bases.

${(3^{4})}^{x} = 3^{2}$

Step 4

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

$3^{4 x} = 3^{2}$

Step 5

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

$4 x = 2$

Step 6

Solve for $x$ .

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

Step 7

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

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Evaluate log base 81 of 9

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