Evaluate log base 27 of 81

$\log_{27} (81)$

Step 1

Rewrite as an equation.

$\log_{27} (81) = x$

Step 2

Rewrite $\log_{27} (81) = x$ 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 $b^{y} = x$ .

$27^{x} = 81$

Step 3

Create expressions in the equation that all have equal bases.

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

Step 4

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

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

Step 5

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

$3 x = 4$

Step 6

Solve for $x$ .

$x = \frac{4}{3}$

Step 7

The variable $x$ is equal to $\frac{4}{3}$ .

$\frac{4}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$

Evaluate log base 27 of 81

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