Solve 25.6=50*0.8^t

$$25.6 = 50 \cdot 0.8 ^ { t }$$

Solve for t

$$t=3$$

Steps Using Definition of Logarithm

Divide both sides by 50.

$$\frac{25.6}{50}=0.8^{t}$$

Expand \frac{25.6}{50} by multiplying both numerator and the denominator by 10.

$$\frac{256}{500}=0.8^{t}$$

Reduce the fraction \frac{256}{500} to lowest terms by extracting and canceling out 4.

$$\frac{64}{125}=0.8^{t}$$

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

$$0.8^{t}=\frac{64}{125}$$

Take the logarithm of both sides of the equation.

$$\log(0.8^{t})=\log(\frac{64}{125})$$

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

$$t\log(0.8)=\log(\frac{64}{125})$$

Divide both sides by \log(0.8).

$$t=\frac{\log(\frac{64}{125})}{\log(0.8)}$$

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

$$t=\log_{0.8}\left(\frac{64}{125}\right)$$