Precalculus Examples

$428^{- x} = 37.5$

Step 1

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

$\ln (428^{- x}) = \ln (37.5)$

Step 2

Expand $\ln (428^{- x})$ by moving $- x$ outside the logarithm.

$- x \ln (428) = \ln (37.5)$

Step 3

Divide each term in $- x \ln (428) = \ln (37.5)$ by $- \ln (428)$ and simplify.

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Step 3.1

Divide each term in $- x \ln (428) = \ln (37.5)$ by $- \ln (428)$ .

$\frac{- x \ln (428)}{- \ln (428)} = \frac{\ln (37.5)}{- \ln (428)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (428)}{\ln (428)} = \frac{\ln (37.5)}{- \ln (428)}$

Step 3.2.2

Cancel the common factor of $\ln (428)$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{x \ln (428)}{\ln (428)} = \frac{\ln (37.5)}{- \ln (428)}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (37.5)}{- \ln (428)}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (37.5)}{\ln (428)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (37.5)}{\ln (428)}$

Decimal Form:

$x = - 0.59816260 \dots$