Solve 2^x-1+5=5^3

$${ 2 }^{ x-1 } +5 = { 5 }^{ 3 }$$

Solve for x

$$x=\log_{2}\left(15\right)+4\approx 7.906890596$$

Steps Using Definition of Logarithm

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

$$2^{x-1}+5=125$$

Subtract 5 from both sides of the equation.

$$2^{x-1}=120$$

Take the logarithm of both sides of the equation.

$$\log(2^{x-1})=\log(120)$$

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

$$\left(x-1\right)\log(2)=\log(120)$$

Divide both sides by \log(2).

$$x-1=\frac{\log(120)}{\log(2)}$$

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

$$x-1=\log_{2}\left(120\right)$$

Add 1 to both sides of the equation.

$$x=\log_{2}\left(120\right)-\left(-1\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(15\right)+4$$

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