Solve 30(2^x-1)=7.5

$$30 ( 2 ^ { x - 1 } ) = 7.5$$

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

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

$$30\times 2^{x-1}=7.5$$

Divide both sides by 30.

$$2^{x-1}=0.25$$

Take the logarithm of both sides of the equation.

$$\log(2^{x-1})=\log(0.25)$$

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

$$\left(x-1\right)\log(2)=\log(0.25)$$

Divide both sides by \log(2).

$$x-1=\frac{\log(0.25)}{\log(2)}$$

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

$$x-1=\log_{2}\left(0.25\right)$$

Add 1 to both sides of the equation.

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

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(2)}-1$$

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