Solve 25=3(0.75)^-2t?

$$25 = 3 ( 0.75 ) ^ { - 2 t } ?$$

Solve for t

$$t = \frac{\log_{\frac{4}{3}} {(\frac{25}{3})}}{2} \approx 3.6850811$$

Steps Using Definition of Logarithm

Divide both sides by 3.

$$\frac{25}{3}=0.75^{-2t}$$

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

$$0.75^{-2t}=\frac{25}{3}$$

Take the logarithm of both sides of the equation.

$$\log(0.75^{-2t})=\log(\frac{25}{3})$$

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

$$-2t\log(0.75)=\log(\frac{25}{3})$$

Divide both sides by \log(0.75).

$$-2t=\frac{\log(\frac{25}{3})}{\log(0.75)}$$

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

$$-2t=\log_{0.75}\left(\frac{25}{3}\right)$$

Divide both sides by -2.

$$t=\frac{\ln(\frac{25}{3})}{-2\ln(\frac{3}{4})}$$