Solve {left(7/8right)}^2x+3=2

$${ \left( \frac{ 7 }{ 8 } \right) }^{ 2x+3 } =2$$

Solve for x

$$x = -\frac{\log_{\frac{7}{8}} {(\frac{343}{1024})}}{2} \approx -4.095446535$$

Steps Using Definition of Logarithm

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

$$\left(\frac{7}{8}\right)^{2x+3}=2$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{7}{8}\right)^{2x+3})=\log(2)$$

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

$$\left(2x+3\right)\log(\frac{7}{8})=\log(2)$$

Divide both sides by \log(\frac{7}{8}).

$$2x+3=\frac{\log(2)}{\log(\frac{7}{8})}$$

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

$$2x+3=\log_{\frac{7}{8}}\left(2\right)$$

Subtract 3 from both sides of the equation.

$$2x=\frac{\ln(2)}{\ln(\frac{7}{8})}-3$$

Divide both sides by 2.

$$x=\frac{\frac{\ln(2)}{\ln(\frac{7}{8})}-3}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{7}{8})}-\frac{\log_{\frac{7}{8}}\left(\frac{343}{1024}\right)}{2}$$

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