Solve 3^5*3^4*3^8=9^x

$$3 ^ { 5 } \cdot 3 ^ { 4 } \cdot 3 ^ { 8 } = 9 ^ { x }$$

Solve for x

$$x = \frac{17}{2} = 8\frac{1}{2} = 8.5$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add 5 and 4 to get 9.

$$3^{9}\times 3^{8}=9^{x}$$

To multiply powers of the same base, add their exponents. Add 9 and 8 to get 17.

$$3^{17}=9^{x}$$

Calculate 3 to the power of 17 and get 129140163.

$$129140163=9^{x}$$

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

$$9^{x}=129140163$$

Take the logarithm of both sides of the equation.

$$\log(9^{x})=\log(129140163)$$

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

$$x\log(9)=\log(129140163)$$

Divide both sides by \log(9).

$$x=\frac{\log(129140163)}{\log(9)}$$

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

$$x=\log_{9}\left(129140163\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(3)}+\frac{17}{2}$$

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