Solve 3^23*3^x*3^19=3^78

$$3 ^ { 23 } \times 3 ^ { x } \times 3 ^ { 19 } = 3 ^ { 78 }$$

Solve for x

$$x=36$$

Steps Using Definition of Logarithm

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

$$109418989131512359209\times 3^{x}=16423203268260658146231467800709255289$$

Divide both sides by 109418989131512359209.

$$3^{x}=150094635296999121$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(150094635296999121)$$

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

$$x\log(3)=\log(150094635296999121)$$

Divide both sides by \log(3).

$$x=\frac{\log(150094635296999121)}{\log(3)}$$

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

$$x=\log_{3}\left(150094635296999121\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}+36$$

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