Solve (3/4)^t=64/27

$$( \frac { 3 } { 4 } ) ^ { t } = \frac { 64 } { 27 }$$

Solve for t

$$t=-3$$

Steps Using Definition of Logarithm

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

$$\left(\frac{3}{4}\right)^{t}=\frac{64}{27}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{4}\right)^{t})=\log(\frac{64}{27})$$

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

$$t\log(\frac{3}{4})=\log(\frac{64}{27})$$

Divide both sides by \log(\frac{3}{4}).

$$t=\frac{\log(\frac{64}{27})}{\log(\frac{3}{4})}$$

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

$$t=\log_{\frac{3}{4}}\left(\frac{64}{27}\right)$$