Solve (3/4)^-9*(3/4)^-7=(3/4)^4x

$$( \frac { 3 } { 4 } ) ^ { - 9 } \times ( \frac { 3 } { 4 } ) ^ { - 7 } = ( \frac { 3 } { 4 } ) ^ { 4 x }$$

Solve for x

$$x=-4$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add -9 and -7 to get -16.

$$\left(\frac{3}{4}\right)^{-16}=\left(\frac{3}{4}\right)^{4x}$$

Calculate \frac{3}{4} to the power of -16 and get \frac{4294967296}{43046721}.

$$\frac{4294967296}{43046721}=\left(\frac{3}{4}\right)^{4x}$$

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

$$\left(\frac{3}{4}\right)^{4x}=\frac{4294967296}{43046721}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{4}\right)^{4x})=\log(\frac{4294967296}{43046721})$$

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

$$4x\log(\frac{3}{4})=\log(\frac{4294967296}{43046721})$$

Divide both sides by \log(\frac{3}{4}).

$$4x=\frac{\log(\frac{4294967296}{43046721})}{\log(\frac{3}{4})}$$

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

$$4x=\log_{\frac{3}{4}}\left(\frac{4294967296}{43046721}\right)$$

Divide both sides by 4.

$$x=-\frac{16}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(\frac{3}{4})}-\frac{\log_{\frac{3}{4}}\left(\frac{6561}{65536}\right)}{2}$$

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