Solve 78*(2)^x=1000

$$78 \cdot ( 2 ) ^ { x } = 1000$$

Solve for x

$$x=\log_{2}\left(\frac{500}{39}\right)\approx 3.680382066$$

Steps Using Definition of Logarithm

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

$$78\times 2^{x}=1000$$

Divide both sides by 78.

$$2^{x}=\frac{500}{39}$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(\frac{500}{39})$$

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

$$x\log(2)=\log(\frac{500}{39})$$

Divide both sides by \log(2).

$$x=\frac{\log(\frac{500}{39})}{\log(2)}$$

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

$$x=\log_{2}\left(\frac{500}{39}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(\frac{500}{39}\right)$$

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