Precalculus Examples

$5^{- \frac{x}{100}} = 2$

Step 1

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

$\ln (5^{- \frac{x}{100}}) = \ln (2)$

Step 2

Expand the left side.

Tap for more steps...

Step 2.1

Expand $\ln (5^{- \frac{x}{100}})$ by moving $- \frac{x}{100}$ outside the logarithm.

$- \frac{x}{100} \ln (5) = \ln (2)$

Step 2.2

Combine $\ln (5)$ and $\frac{x}{100}$ .

$- \frac{\ln (5) x}{100} = \ln (2)$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Reorder factors in $- \frac{\ln (5) x}{100}$ .

$- \frac{x \ln (5)}{100} = \ln (2)$

Step 4

Multiply both sides of the equation by $- \frac{100}{\ln (5)}$ .

$- \frac{100}{\ln (5)} (- \frac{x \ln (5)}{100}) = - \frac{100}{\ln (5)} \ln (2)$

Step 5

Simplify both sides of the equation.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Simplify $- \frac{100}{\ln (5)} (- \frac{x \ln (5)}{100})$ .

Tap for more steps...

Step 5.1.1.1

Cancel the common factor of $100$ .

Tap for more steps...

Step 5.1.1.1.1

Move the leading negative in $- \frac{100}{\ln (5)}$ into the numerator.

$\frac{- 100}{\ln (5)} (- \frac{x \ln (5)}{100}) = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.1.2

Move the leading negative in $- \frac{x \ln (5)}{100}$ into the numerator.

$\frac{- 100}{\ln (5)} \cdot \frac{- x \ln (5)}{100} = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.1.3

Factor $100$ out of $- 100$ .

$\frac{100 (- 1)}{\ln (5)} \cdot \frac{- x \ln (5)}{100} = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{100 \cdot - 1}{\ln (5)} \cdot \frac{- x \ln (5)}{100} = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{\ln (5)} (- x \ln (5)) = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.2

Cancel the common factor of $\ln (5)$ .

Tap for more steps...

Step 5.1.1.2.1

Factor $\ln (5)$ out of $- x \ln (5)$ .

$\frac{- 1}{\ln (5)} (\ln (5) (- x)) = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{\ln (5)} (\ln (5) (- x)) = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.2.3

Rewrite the expression.

$- - x = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.3

Multiply.

Tap for more steps...

Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{100}{\ln (5)} \ln (2)$

Step 5.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{100}{\ln (5)} \ln (2)$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $- \frac{100}{\ln (5)} \ln (2)$ .

Tap for more steps...

Step 5.2.1.1

Combine $\ln (2)$ and $\frac{100}{\ln (5)}$ .

$x = - \frac{\ln (2) \cdot 100}{\ln (5)}$

Step 5.2.1.2

Move $100$ to the left of $\ln (2)$ .

$x = - \frac{100 \ln (2)}{\ln (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{100 \ln (2)}{\ln (5)}$

Decimal Form:

$x = - 43.06765580 \dots$