Solve 1.03^x=1.5

$$1.03 ^ { x } = 1.5$$

Solve for x

$$x=\log_{1.03}\left(1.5\right)\approx 13.717237417$$

Steps Using Definition of Logarithm

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

$$1.03^{x}=1.5$$

Take the logarithm of both sides of the equation.

$$\log(1.03^{x})=\log(1.5)$$

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

$$x\log(1.03)=\log(1.5)$$

Divide both sides by \log(1.03).

$$x=\frac{\log(1.5)}{\log(1.03)}$$

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

$$x=\log_{1.03}\left(1.5\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.03)}+\log_{1.03}\left(1.5\right)$$

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