Solve 0.913^t=0.142.quadt=?

$$0.913 ^ { t } = 0.142 . \quad t = ?$$

Solve for t

$$t=\log_{0.913}\left(0.142\right)\approx 21.445189223$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$0.913^{t}=0.142$$

Take the logarithm of both sides of the equation.

$$\log(0.913^{t})=\log(0.142)$$

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

$$t\log(0.913)=\log(0.142)$$

Divide both sides by \log(0.913).

$$t=\frac{\log(0.142)}{\log(0.913)}$$

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

$$t=\log_{0.913}\left(0.142\right)$$