Solve 12(3/7)^-frac{1{2}x}=28

$$12 ( \frac { 3 } { 7 } ) ^ { - \frac { 1 } { 2 } x } = 28$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

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

$$12\times \left(\frac{3}{7}\right)^{-\frac{1}{2}x}=28$$

Divide both sides by 12.

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

Take the logarithm of both sides of the equation.

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

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

$$-\frac{1}{2}x\log(\frac{3}{7})=\log(\frac{7}{3})$$

Divide both sides by \log(\frac{3}{7}).

$$-\frac{1}{2}x=\frac{\log(\frac{7}{3})}{\log(\frac{3}{7})}$$

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

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

Multiply both sides by -2.

$$x=-\frac{1}{-\frac{1}{2}}$$

Solve for x (complex solution)

$$x=-\frac{4\pi n_{1}i}{\ln(\frac{3}{7})}+2$$

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