Solve 5^x-1=15625/243

$${ 5 }^{ x-1 } = \frac{ 15625 }{ 243 }$$

Solve for x

$$x=-5\log_{5}\left(3\right)+7\approx 3.586969028$$

Steps Using Definition of Logarithm

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

$$5^{x-1}=\frac{15625}{243}$$

Take the logarithm of both sides of the equation.

$$\log(5^{x-1})=\log(\frac{15625}{243})$$

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

$$\left(x-1\right)\log(5)=\log(\frac{15625}{243})$$

Divide both sides by \log(5).

$$x-1=\frac{\log(\frac{15625}{243})}{\log(5)}$$

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

$$x-1=\log_{5}\left(\frac{15625}{243}\right)$$

Add 1 to both sides of the equation.

$$x=-5\log_{5}\left(3\right)+6-\left(-1\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(5)}-5\log_{5}\left(3\right)+7$$

$$n_{1}\in \mathrm{Z}$$