Solve 6=96(0.88)^x

$$6 = 96 ( 0.88 ) ^ { x }$$

Solve for x

$$x=-4\log_{0.88}\left(2\right)\approx 21.689083918$$

Steps Using Definition of Logarithm

Divide both sides by 96.

$$\frac{6}{96}=0.88^{x}$$

Reduce the fraction \frac{6}{96} to lowest terms by extracting and canceling out 6.

$$\frac{1}{16}=0.88^{x}$$

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

$$0.88^{x}=\frac{1}{16}$$

Take the logarithm of both sides of the equation.

$$\log(0.88^{x})=\log(\frac{1}{16})$$

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

$$x\log(0.88)=\log(\frac{1}{16})$$

Divide both sides by \log(0.88).

$$x=\frac{\log(\frac{1}{16})}{\log(0.88)}$$

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

$$x=\log_{0.88}\left(\frac{1}{16}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.88)}-4\log_{0.88}\left(2\right)$$

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