Solve 10.3^x-3=810

$${ 10.3 }^{ x-3 } =810$$

Solve for x

$$x=-\log_{10.3}\left(\frac{100}{88510887}\right)\approx 5.87162137$$

Steps Using Definition of Logarithm

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

$$10.3^{x-3}=810$$

Take the logarithm of both sides of the equation.

$$\log(10.3^{x-3})=\log(810)$$

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

$$\left(x-3\right)\log(10.3)=\log(810)$$

Divide both sides by \log(10.3).

$$x-3=\frac{\log(810)}{\log(10.3)}$$

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

$$x-3=\log_{10.3}\left(810\right)$$

Add 3 to both sides of the equation.

$$x=\frac{\ln(810)}{\ln(\frac{103}{10})}-\left(-3\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(10.3)}-\log_{10.3}\left(\frac{100}{88510887}\right)$$

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