Solve 10^-5=(2/5)^t

$$10 ^ { - 5 } = ( \frac { 2 } { 5 } ) ^ { t }$$

Solve for t

$$t=-5\log_{\frac{2}{5}}\left(10\right)\approx 12.564707974$$

Steps Using Definition of Logarithm

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$\frac{1}{100000}=\left(\frac{2}{5}\right)^{t}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{2}{5}\right)^{t}=\frac{1}{100000}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{5}\right)^{t})=\log(\frac{1}{100000})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$t\log(\frac{2}{5})=\log(\frac{1}{100000})$$

Divide both sides by \log(\frac{2}{5}).

$$t=\frac{\log(\frac{1}{100000})}{\log(\frac{2}{5})}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$t=\log_{\frac{2}{5}}\left(\frac{1}{100000}\right)$$