Solve 160-20-20{left(frac{7/8right)}^x}{1/8}=0.5

$$160- \frac{ 20-20 { \left( \frac{ 7 }{ 8 } \right) }^{ x } }{ \frac{ 1 }{ 8 } } =0.5$$

Solve for x

$$x=-\log_{0.875}\left(320\right)\approx 43.198238874$$

Steps Using Definition of Logarithm

Divide each term of 20-20\times \left(\frac{7}{8}\right)^{x} by \frac{1}{8} to get \frac{20}{\frac{1}{8}}-\frac{20\times \left(\frac{7}{8}\right)^{x}}{\frac{1}{8}}.

$$160-\left(\frac{20}{\frac{1}{8}}-\frac{20\times \left(\frac{7}{8}\right)^{x}}{\frac{1}{8}}\right)=0.5$$

Divide 20 by \frac{1}{8} by multiplying 20 by the reciprocal of \frac{1}{8}.

$$160-\left(20\times 8-\frac{20\times \left(\frac{7}{8}\right)^{x}}{\frac{1}{8}}\right)=0.5$$

Multiply 20 and 8 to get 160.

$$160-\left(160-\frac{20\times \left(\frac{7}{8}\right)^{x}}{\frac{1}{8}}\right)=0.5$$

Divide 20\times \left(\frac{7}{8}\right)^{x} by \frac{1}{8} to get 160\times \left(\frac{7}{8}\right)^{x}.

$$160-\left(160-160\times \left(\frac{7}{8}\right)^{x}\right)=0.5$$

To find the opposite of 160-160\times \left(\frac{7}{8}\right)^{x}, find the opposite of each term.

$$160-160-\left(-160\times \left(\frac{7}{8}\right)^{x}\right)=0.5$$

Subtract 160 from 160 to get 0.

$$-\left(-160\times \left(\frac{7}{8}\right)^{x}\right)=0.5$$

Divide both sides by -1.

$$-160\times \left(\frac{7}{8}\right)^{x}=\frac{0.5}{-1}$$

Expand \frac{0.5}{-1} by multiplying both numerator and the denominator by 10.

$$-160\times \left(\frac{7}{8}\right)^{x}=\frac{5}{-10}$$

Reduce the fraction \frac{5}{-10} to lowest terms by extracting and canceling out 5.

$$-160\times \left(\frac{7}{8}\right)^{x}=-\frac{1}{2}$$

Divide both sides by -1.

$$160\times \left(\frac{7}{8}\right)^{x}=\frac{-\frac{1}{2}}{-1}$$

Express \frac{-\frac{1}{2}}{-1} as a single fraction.

$$160\times \left(\frac{7}{8}\right)^{x}=\frac{-1}{2\left(-1\right)}$$

Cancel out -1 in both numerator and denominator.

$$160\times \left(\frac{7}{8}\right)^{x}=\frac{1}{2}$$

Divide both sides by 160.

$$\left(\frac{7}{8}\right)^{x}=\frac{\frac{1}{2}}{160}$$

Express \frac{\frac{1}{2}}{160} as a single fraction.

$$\left(\frac{7}{8}\right)^{x}=\frac{1}{2\times 160}$$

Multiply 2 and 160 to get 320.

$$\left(\frac{7}{8}\right)^{x}=\frac{1}{320}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{7}{8}\right)^{x})=\log(\frac{1}{320})$$

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

$$x\log(\frac{7}{8})=\log(\frac{1}{320})$$

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

$$x=\frac{\log(\frac{1}{320})}{\log(\frac{7}{8})}$$

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

$$x=\log_{\frac{7}{8}}\left(\frac{1}{320}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.875)}-\log_{0.875}\left(320\right)$$

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