Solve {left(4/3right)}^-4{left(4/3right)}^-5={left(4/3right)}^-3x

$${ \left( \frac{ 4 }{ 3 } \right) }^{ -4 } { \left( \frac{ 4 }{ 3 } \right) }^{ -5 } = { \left( \frac{ 4 }{ 3 } \right) }^{ -3x }$$

Solve for x

$$x=3$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add -4 and -5 to get -9.

$$\left(\frac{4}{3}\right)^{-9}=\left(\frac{4}{3}\right)^{-3x}$$

Calculate \frac{4}{3} to the power of -9 and get \frac{19683}{262144}.

$$\frac{19683}{262144}=\left(\frac{4}{3}\right)^{-3x}$$

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

$$\left(\frac{4}{3}\right)^{-3x}=\frac{19683}{262144}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{4}{3}\right)^{-3x})=\log(\frac{19683}{262144})$$

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

$$-3x\log(\frac{4}{3})=\log(\frac{19683}{262144})$$

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

$$-3x=\frac{\log(\frac{19683}{262144})}{\log(\frac{4}{3})}$$

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

$$-3x=\log_{\frac{4}{3}}\left(\frac{19683}{262144}\right)$$

Divide both sides by -3.

$$x=-\frac{9}{-3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(\frac{3}{4})}-\frac{\log_{\frac{3}{4}}\left(\frac{262144}{19683}\right)}{3}$$

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