Solve 1.1^x-1=5.5div1.8

$${ 1.1 }^{ x } -1 = 5.5 \div 1.8$$

Solve for x

$$x=\log_{1.1}\left(\frac{73}{18}\right)\approx 14.68980214$$

Steps Using Definition of Logarithm

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

$$1.1^{x}-1=\frac{55}{18}$$

Add 1 to both sides of the equation.

$$1.1^{x}=\frac{73}{18}$$

Take the logarithm of both sides of the equation.

$$\log(1.1^{x})=\log(\frac{73}{18})$$

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

$$x\log(1.1)=\log(\frac{73}{18})$$

Divide both sides by \log(1.1).

$$x=\frac{\log(\frac{73}{18})}{\log(1.1)}$$

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

$$x=\log_{1.1}\left(\frac{73}{18}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.1)}+\log_{1.1}\left(\frac{73}{18}\right)$$

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