Solve 3.5^x-1=75

$$3.5 ^ { x - 1 } = 75$$

Solve for x

$$x=\log_{3.5}\left(262.5\right)\approx 4.446372715$$

Steps Using Definition of Logarithm

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

$$3.5^{x-1}=75$$

Take the logarithm of both sides of the equation.

$$\log(3.5^{x-1})=\log(75)$$

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

$$\left(x-1\right)\log(3.5)=\log(75)$$

Divide both sides by \log(3.5).

$$x-1=\frac{\log(75)}{\log(3.5)}$$

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

$$x-1=\log_{3.5}\left(75\right)$$

Add 1 to both sides of the equation.

$$x=\frac{\ln(75)}{\ln(\frac{7}{2})}-\left(-1\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(3.5)}-\log_{3.5}\left(\frac{2}{525}\right)$$

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