Solve 6^-5*6^2x=6^7

$${ 6 }^{ -5 } \times { 6 }^{ 2x } = { 6 }^{ 7 }$$

Solve for x

$$x=6$$

Steps Using Definition of Logarithm

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

$$\frac{1}{7776}\times 6^{2x}=279936$$

Multiply both sides by 7776.

$$6^{2x}=2176782336$$

Take the logarithm of both sides of the equation.

$$\log(6^{2x})=\log(2176782336)$$

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

$$2x\log(6)=\log(2176782336)$$

Divide both sides by \log(6).

$$2x=\frac{\log(2176782336)}{\log(6)}$$

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

$$2x=\log_{6}\left(2176782336\right)$$

Divide both sides by 2.

$$x=\frac{12}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(6)}+6$$

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