Solve 161/5*2^x=83/4

$$16 \frac { 1 } { 5 } \times 2 ^ { x } = 8 \frac { 3 } { 4 }$$

Solve for x

$$x=\log_{2}\left(\frac{175}{324}\right)\approx -0.888638891$$

Steps Using Definition of Logarithm

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

$$\frac{81}{5}\times 2^{x}=\frac{35}{4}$$

Divide both sides of the equation by \frac{81}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$2^{x}=\frac{175}{324}$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(\frac{175}{324})$$

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

$$x\log(2)=\log(\frac{175}{324})$$

Divide both sides by \log(2).

$$x=\frac{\log(\frac{175}{324})}{\log(2)}$$

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

$$x=\log_{2}\left(\frac{175}{324}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(\frac{175}{324}\right)$$

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