Solve 3^n-1=131/8

$${ 3 }^{ n-1 } = 13 \frac{ 1 }{ 8 }$$

Solve for n

$$n=\log_{3}\left(\frac{35}{8}\right)+2\approx 3.343428009$$

Steps Using Definition of Logarithm

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

$$3^{n-1}=\frac{105}{8}$$

Take the logarithm of both sides of the equation.

$$\log(3^{n-1})=\log(\frac{105}{8})$$

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

$$\left(n-1\right)\log(3)=\log(\frac{105}{8})$$

Divide both sides by \log(3).

$$n-1=\frac{\log(\frac{105}{8})}{\log(3)}$$

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

$$n-1=\log_{3}\left(\frac{105}{8}\right)$$

Add 1 to both sides of the equation.

$$n=\frac{\ln(\frac{105}{8})}{\ln(3)}-\left(-1\right)$$