Solve (1/2)^-5*(1/3)^7=(1/3)^2x-1

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

Solve for x

$$x=-\frac{5\log_{3}\left(2\right)}{2}+4\approx 2.422675616$$

Steps Using Definition of Logarithm

Calculate \frac{1}{2} to the power of -5 and get 32.

$$32\times \left(\frac{1}{3}\right)^{7}=\left(\frac{1}{3}\right)^{2x-1}$$

Calculate \frac{1}{3} to the power of 7 and get \frac{1}{2187}.

$$32\times \frac{1}{2187}=\left(\frac{1}{3}\right)^{2x-1}$$

Multiply 32 and \frac{1}{2187} to get \frac{32}{2187}.

$$\frac{32}{2187}=\left(\frac{1}{3}\right)^{2x-1}$$

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

$$\left(\frac{1}{3}\right)^{2x-1}=\frac{32}{2187}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{2x-1})=\log(\frac{32}{2187})$$

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

$$\left(2x-1\right)\log(\frac{1}{3})=\log(\frac{32}{2187})$$

Divide both sides by \log(\frac{1}{3}).

$$2x-1=\frac{\log(\frac{32}{2187})}{\log(\frac{1}{3})}$$

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

$$2x-1=\log_{\frac{1}{3}}\left(\frac{32}{2187}\right)$$

Add 1 to both sides of the equation.

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

Divide both sides by 2.

$$x=\frac{-5\log_{3}\left(2\right)+8}{2}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{\ln(3)}-\frac{5\log_{3}\left(2\right)}{2}+4$$

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