Solve 75.75=100*0.99988^x

$$75.75=100 \times { 0.99988 }^{ x }$$

Solve for x

$$x=\log_{0.99988}\left(0.7575\right)\approx 2314.29231134$$

Steps Using Definition of Logarithm

Divide both sides by 100.

$$\frac{75.75}{100}=0.99988^{x}$$

Expand \frac{75.75}{100} by multiplying both numerator and the denominator by 100.

$$\frac{7575}{10000}=0.99988^{x}$$

Reduce the fraction \frac{7575}{10000} to lowest terms by extracting and canceling out 25.

$$\frac{303}{400}=0.99988^{x}$$

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

$$0.99988^{x}=\frac{303}{400}$$

Take the logarithm of both sides of the equation.

$$\log(0.99988^{x})=\log(\frac{303}{400})$$

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

$$x\log(0.99988)=\log(\frac{303}{400})$$

Divide both sides by \log(0.99988).

$$x=\frac{\log(\frac{303}{400})}{\log(0.99988)}$$

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

$$x=\log_{0.99988}\left(\frac{303}{400}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.99988)}+\log_{0.99988}\left(0.7575\right)$$

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