Solve 6^x*2=5

$$6 ^ { x } \cdot 2 = 5$$

Solve for x

$$x=\log_{6}\left(\frac{5}{2}\right)\approx 0.511391594$$

Steps Using Definition of Logarithm

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

$$2\times 6^{x}=5$$

Divide both sides by 2.

$$6^{x}=\frac{5}{2}$$

Take the logarithm of both sides of the equation.

$$\log(6^{x})=\log(\frac{5}{2})$$

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

$$x\log(6)=\log(\frac{5}{2})$$

Divide both sides by \log(6).

$$x=\frac{\log(\frac{5}{2})}{\log(6)}$$

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

$$x=\log_{6}\left(\frac{5}{2}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(6)}+\log_{6}\left(\frac{5}{2}\right)$$

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