Solve 7.4^x-3=32.5

$${ 7.4 }^{ x-3 } =32.5$$

Solve for x

$$x=-\log_{7.4}\left(\frac{50}{658489}\right)\approx 4.739332938$$

Steps Using Definition of Logarithm

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

$$7.4^{x-3}=32.5$$

Take the logarithm of both sides of the equation.

$$\log(7.4^{x-3})=\log(32.5)$$

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

$$\left(x-3\right)\log(7.4)=\log(32.5)$$

Divide both sides by \log(7.4).

$$x-3=\frac{\log(32.5)}{\log(7.4)}$$

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

$$x-3=\log_{7.4}\left(32.5\right)$$

Add 3 to both sides of the equation.

$$x=\frac{\ln(\frac{65}{2})}{\ln(\frac{37}{5})}-\left(-3\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(7.4)}-\log_{7.4}\left(\frac{50}{658489}\right)$$

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