Trigonometry Examples

$25 (125^{x}) = 1$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (25 \cdot 125^{x}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (25 \cdot 125^{x})$ as $\ln (25) + \ln (125^{x})$ .

$\ln (25) + \ln (125^{x}) = \ln (1)$

Step 2.2

Expand $\ln (125^{x})$ by moving $x$ outside the logarithm.

$\ln (25) + x \ln (125) = \ln (1)$

Step 3

Simplify the right side.

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Step 3.1

The natural logarithm of $1$ is $0$ .

$\ln (25) + x \ln (125) = 0$

Step 4

Reorder $\ln (25)$ and $x \ln (125)$ .

$x \ln (125) + \ln (25) = 0$

Step 5

Subtract $\ln (25)$ from both sides of the equation.

$x \ln (125) = - \ln (25)$

Step 6

Divide each term in $x \ln (125) = - \ln (25)$ by $\ln (125)$ and simplify.

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Step 6.1

Divide each term in $x \ln (125) = - \ln (25)$ by $\ln (125)$ .

$\frac{x \ln (125)}{\ln (125)} = \frac{- \ln (25)}{\ln (125)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (125)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x \ln (125)}{\ln (125)} = \frac{- \ln (25)}{\ln (125)}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (25)}{\ln (125)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (25)}{\ln (125)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (25)}{\ln (125)}$

Decimal Form:

$x = - 0.\overline{6}$