Solve {left(3/5right)}^3{left(3/5right)}^-6={left(3/5right)}^2x-1

$${ \left( \frac{ 3 }{ 5 } \right) }^{ 3 } { \left( \frac{ 3 }{ 5 } \right) }^{ -6 } = { \left( \frac{ 3 }{ 5 } \right) }^{ 2x-1 }$$

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add 3 and -6 to get -3.

$$\left(\frac{3}{5}\right)^{-3}=\left(\frac{3}{5}\right)^{2x-1}$$

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

$$\frac{125}{27}=\left(\frac{3}{5}\right)^{2x-1}$$

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

$$\left(\frac{3}{5}\right)^{2x-1}=\frac{125}{27}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{5}\right)^{2x-1})=\log(\frac{125}{27})$$

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

$$\left(2x-1\right)\log(\frac{3}{5})=\log(\frac{125}{27})$$

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

$$2x-1=\frac{\log(\frac{125}{27})}{\log(\frac{3}{5})}$$

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

$$2x-1=\log_{\frac{3}{5}}\left(\frac{125}{27}\right)$$

Add 1 to both sides of the equation.

$$2x=-3-\left(-1\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

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

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