Solve (3/5)^-3*(5/3)^xdiv(3/5)^-2=(3/5)^-3

$$( \frac { 3 } { 5 } ) ^ { - 3 } \times ( \frac { 5 } { 3 } ) ^ { x } \div ( \frac { 3 } { 5 } ) ^ { - 2 } = ( \frac { 3 } { 5 } ) ^ { - 3 }$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(\frac{5}{3}\right)^{x}}{\left(\frac{3}{5}\right)^{1}}=\left(\frac{3}{5}\right)^{-3}$$

Calculate \frac{3}{5} to the power of 1 and get \frac{3}{5}.

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

Calculate \frac{3}{5} to the power of -3 and get \frac{125}{27}.

$$\frac{\left(\frac{5}{3}\right)^{x}}{\frac{3}{5}}=\frac{125}{27}$$

Multiply both sides by \frac{3}{5}.

$$\left(\frac{5}{3}\right)^{x}=\frac{125}{27}\times \frac{3}{5}$$

Multiply \frac{125}{27} and \frac{3}{5} to get \frac{25}{9}.

$$\left(\frac{5}{3}\right)^{x}=\frac{25}{9}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{3}\right)^{x})=\log(\frac{25}{9})$$

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

$$x\log(\frac{5}{3})=\log(\frac{25}{9})$$

Divide both sides by \log(\frac{5}{3}).

$$x=\frac{\log(\frac{25}{9})}{\log(\frac{5}{3})}$$

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

$$x=\log_{\frac{5}{3}}\left(\frac{25}{9}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{5}{3})}+2$$

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