Algebra Examples

$\log_{(\frac{3}{5})} (x) = - 3$

Step 1

Rewrite $\log_{\frac{3}{5}} (x) = - 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(\frac{3}{5})}^{- 3} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = {(\frac{3}{5})}^{- 3}$ .

$x = {(\frac{3}{5})}^{- 3}$

Step 2.2

Simplify ${(\frac{3}{5})}^{- 3}$ .

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Step 2.2.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$x = {(\frac{5}{3})}^{3}$

Step 2.2.2

Apply the product rule to $\frac{5}{3}$ .

$x = \frac{5^{3}}{3^{3}}$

Step 2.2.3

Raise $5$ to the power of $3$ .

$x = \frac{125}{3^{3}}$

Step 2.2.4

Raise $3$ to the power of $3$ .

$x = \frac{125}{27}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{125}{27}$

Decimal Form:

$x = 4.\overline{629}$

Mixed Number Form:

$x = 4 \frac{17}{27}$