Solve 12(2.5)^x-1=75

$$12 ( 2.5 ) ^ { x - 1 } = 75$$

Solve for x

$$x=3$$

Steps Using Definition of Logarithm

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

$$12\times 2.5^{x-1}=75$$

Divide both sides by 12.

$$2.5^{x-1}=\frac{25}{4}$$

Take the logarithm of both sides of the equation.

$$\log(2.5^{x-1})=\log(\frac{25}{4})$$

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

$$\left(x-1\right)\log(2.5)=\log(\frac{25}{4})$$

Divide both sides by \log(2.5).

$$x-1=\frac{\log(\frac{25}{4})}{\log(2.5)}$$

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

$$x-1=\log_{2.5}\left(\frac{25}{4}\right)$$

Add 1 to both sides of the equation.

$$x=2-\left(-1\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(2.5)}-\log_{2.5}\left(0.064\right)$$

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