Solve for n 10^(-9n)=24

$10^{- 9 n} = 24$

Step 1

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{- 9 n}) = \log (24)$

Step 2

Expand the left side.

Expand $\log (10^{- 9 n})$ by moving $- 9 n$ outside the logarithm.

$- 9 n \log (10) = \log (24)$

Logarithm base $10$ of $10$ is $1$ .

$- 9 n \cdot 1 = \log (24)$

Multiply $- 9$ by $1$ .

$- 9 n = \log (24)$

Step 3

Divide each term in $- 9 n = \log (24)$ by $- 9$ and simplify.

Divide each term in $- 9 n = \log (24)$ by $- 9$ .

$\frac{- 9 n}{- 9} = \frac{\log (24)}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 n}{- 9} = \frac{\log (24)}{- 9}$

Divide $n$ by $1$ .

$n = \frac{\log (24)}{- 9}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{\log (24)}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\log (24)}{9}$

Decimal Form:

$n = - 0.15335680 \dots$