Solve 80=2^x*3^2*5

$$80 = 2 ^ { x } \times 3 ^ { 2 } \times 5$$

Solve for x

$$x=-2\log_{2}\left(3\right)+4\approx 0.830074999$$

Steps Using Definition of Logarithm

Calculate 3 to the power of 2 and get 9.

$$80=2^{x}\times 9\times 5$$

Multiply 9 and 5 to get 45.

$$80=2^{x}\times 45$$

Swap sides so that all variable terms are on the left hand side.

$$2^{x}\times 45=80$$

Divide both sides by 45.

$$2^{x}=\frac{80}{45}$$

Reduce the fraction \frac{80}{45} to lowest terms by extracting and canceling out 5.

$$2^{x}=\frac{16}{9}$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(\frac{16}{9})$$

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

$$x\log(2)=\log(\frac{16}{9})$$

Divide both sides by \log(2).

$$x=\frac{\log(\frac{16}{9})}{\log(2)}$$

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

$$x=\log_{2}\left(\frac{16}{9}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}-2\log_{2}\left(3\right)+4$$

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