Solve (3/5)^4*(3/5)^3=(3/5)^2x-1

$$( \frac { 3 } { 5 } ) ^ { 4 } \times ( \frac { 3 } { 5 } ) ^ { 3 } = ( \frac { 3 } { 5 } ) ^ { 2 x - 1 }$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\left(\frac{3}{5}\right)^{7}=\left(\frac{3}{5}\right)^{2x-1}$$

Calculate \frac{3}{5} to the power of 7 and get \frac{2187}{78125}.

$$\frac{2187}{78125}=\left(\frac{3}{5}\right)^{2x-1}$$

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

$$\left(\frac{3}{5}\right)^{2x-1}=\frac{2187}{78125}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{5}\right)^{2x-1})=\log(\frac{2187}{78125})$$

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

$$\left(2x-1\right)\log(\frac{3}{5})=\log(\frac{2187}{78125})$$

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

$$2x-1=\frac{\log(\frac{2187}{78125})}{\log(\frac{3}{5})}$$

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

$$2x-1=\log_{\frac{3}{5}}\left(\frac{2187}{78125}\right)$$

Add 1 to both sides of the equation.

$$2x=7-\left(-1\right)$$

Divide both sides by 2.

$$x=\frac{8}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{3}{5})}-\log_{\frac{3}{5}}\left(\frac{625}{81}\right)$$

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