Precalculus Examples

$125 = 25 {(\frac{1}{5})}^{2 x}$

Step 1

Simplify both sides of the equation.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$125 = 5^{2} {(\frac{1}{5})}^{2 x}$

Step 1.2

Rewrite $\frac{1}{5}$ as $5^{- 1}$ .

$125 = 5^{2} {(5^{- 1})}^{2 x}$

Step 1.3

Raise $5$ to the power of $1$ .

$125 = 5^{2} {({(5^{1})}^{- 1})}^{2 x}$

Step 1.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$125 = 5^{2} {(5^{1 \cdot - 1})}^{2 x}$

Step 1.5

Multiply $- 1$ by $1$ .

$125 = 5^{2} {(5^{- 1})}^{2 x}$

Step 1.6

Multiply the exponents in ${(5^{- 1})}^{2 x}$ .

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Step 1.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$125 = 5^{2} \cdot 5^{- (2 x)}$

Step 1.6.2

Multiply $2$ by $- 1$ .

$125 = 5^{2} \cdot 5^{- 2 x}$

Step 1.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$125 = 5^{2 - 2 x}$

Step 2

Convert the exponential equation to a logarithmic equation using the logarithm base $(5)$ of the left side $(125)$ equals the exponent $(2 - 2 x)$ .

$\log_{5} (125) = 2 - 2 x$