Solve (286/92)^n-1=2

$$( \frac { 286 } { 92 } ) ^ { n - 1 } = 2$$

Solve for n

$$n=-\log_{\frac{143}{46}}\left(\frac{23}{143}\right)\approx 1.611131374$$

Steps Using Definition of Logarithm

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

$$\left(\frac{143}{46}\right)^{n-1}=2$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{143}{46}\right)^{n-1})=\log(2)$$

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

$$\left(n-1\right)\log(\frac{143}{46})=\log(2)$$

Divide both sides by \log(\frac{143}{46}).

$$n-1=\frac{\log(2)}{\log(\frac{143}{46})}$$

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

$$n-1=\log_{\frac{143}{46}}\left(2\right)$$

Add 1 to both sides of the equation.

$$n=\frac{\ln(2)}{\ln(\frac{143}{46})}-\left(-1\right)$$