Evaluate log base 27 of 1/3

$\log_{27} (\frac{1}{3})$

Step 1

Rewrite as an equation.

$\log_{27} (\frac{1}{3}) = x$

Step 2

Rewrite $\log_{27} (\frac{1}{3}) = x$ 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 $b^{y} = x$ .

$27^{x} = \frac{1}{3}$

Step 3

Create expressions in the equation that all have equal bases.

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

Step 4

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

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

Step 5

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

$3 x = - 1$

Step 6

Solve for $x$ .

$x = - \frac{1}{3}$

Step 7

The variable $x$ is equal to $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$

Evaluate log base 27 of 1/3

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