Solve 5(6^x)=10(12-6^x)

$$5 ( 6 ^ { x } ) = 10 ( 12 - 6 ^ { x } )$$

Solve for x

$$x=3\log_{6}\left(2\right)\approx 1.160558422$$

Steps Using Definition of Logarithm

Subtract 10\left(12-6^{x}\right) from both sides.

$$5\times 6^{x}-10\left(12-6^{x}\right)=0$$

Use the distributive property to multiply -10 by 12-6^{x}.

$$5\times 6^{x}-120+10\times 6^{x}=0$$

Combine 5\times 6^{x} and 10\times 6^{x} to get 15\times 6^{x}.

$$15\times 6^{x}-120=0$$

Add 120 to both sides of the equation.

$$15\times 6^{x}=120$$

Divide both sides by 15.

$$6^{x}=8$$

Take the logarithm of both sides of the equation.

$$\log(6^{x})=\log(8)$$

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

$$x\log(6)=\log(8)$$

Divide both sides by \log(6).

$$x=\frac{\log(8)}{\log(6)}$$

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

$$x=\log_{6}\left(8\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(6)}+3\log_{6}\left(2\right)$$

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