Solve (5/3)^2x-1=(27/125)^-2

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

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$\left(\frac{5}{3}\right)^{2x-1}=\frac{15625}{729}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{3}\right)^{2x-1})=\log(\frac{15625}{729})$$

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

$$\left(2x-1\right)\log(\frac{5}{3})=\log(\frac{15625}{729})$$

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

$$2x-1=\frac{\log(\frac{15625}{729})}{\log(\frac{5}{3})}$$

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

$$2x-1=\log_{\frac{5}{3}}\left(\frac{15625}{729}\right)$$

Add 1 to both sides of the equation.

$$2x=6-\left(-1\right)$$

Divide both sides by 2.

$$x=\frac{7}{2}$$

Solve for x (complex solution)

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

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