Algebra Examples

$\log_{125} (25)$

Step 1

Rewrite as an equation.

$\log_{125} (25) = x$

Step 2

Rewrite $\log_{125} (25) = x$ 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 $b^{y} = x$ .

$125^{x} = 25$

Step 3

Create expressions in the equation that all have equal bases.

${(5^{3})}^{x} = 5^{2}$

Step 4

Rewrite ${(5^{3})}^{x}$ as $5^{3 x}$ .

$5^{3 x} = 5^{2}$

Step 5

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

$3 x = 2$

Step 6

Solve for $x$ .

$x = \frac{2}{3}$

Step 7

The variable $x$ is equal to $\frac{2}{3}$ .

$\frac{2}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$