Solve (2/7)^-3*(2/7)^-11=(2/7)^x

$$( \frac { 2 } { 7 } ) ^ { - 3 } \times ( \frac { 2 } { 7 } ) ^ { - 11 } = ( \frac { 2 } { 7 } ) ^ { x }$$

Solve for x

$$x=-14$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add -3 and -11 to get -14.

$$\left(\frac{2}{7}\right)^{-14}=\left(\frac{2}{7}\right)^{x}$$

Calculate \frac{2}{7} to the power of -14 and get \frac{678223072849}{16384}.

$$\frac{678223072849}{16384}=\left(\frac{2}{7}\right)^{x}$$

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

$$\left(\frac{2}{7}\right)^{x}=\frac{678223072849}{16384}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{7}\right)^{x})=\log(\frac{678223072849}{16384})$$

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

$$x\log(\frac{2}{7})=\log(\frac{678223072849}{16384})$$

Divide both sides by \log(\frac{2}{7}).

$$x=\frac{\log(\frac{678223072849}{16384})}{\log(\frac{2}{7})}$$

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

$$x=\log_{\frac{2}{7}}\left(\frac{678223072849}{16384}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{2}{7})}+2\log_{\frac{2}{7}}\left(\frac{823543}{128}\right)$$

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