Solve 81^x-4=3^10

$${ 81 }^{ x-4 } = { 3 }^{ 10 }$$

Solve for x

$$x = \frac{13}{2} = 6\frac{1}{2} = 6.5$$

Steps Using Definition of Logarithm

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

$$81^{x-4}=59049$$

Take the logarithm of both sides of the equation.

$$\log(81^{x-4})=\log(59049)$$

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

$$\left(x-4\right)\log(81)=\log(59049)$$

Divide both sides by \log(81).

$$x-4=\frac{\log(59049)}{\log(81)}$$

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

$$x-4=\log_{81}\left(59049\right)$$

Add 4 to both sides of the equation.

$$x=\frac{5}{2}-\left(-4\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(3)}+\frac{13}{2}$$

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