Solve e^-frac{50(frac{21*10^-3{2})}{0.5125}x}=65-100/25-100

$$e ^ { - \frac { 50 ( \frac { 21 \times 10 ^ { - 3 } } { 2 } ) } { 0.5125 } x } = \frac { 65 - 100 } { 25 - 100 }$$

Solve for x

$$x=\frac{41\ln(\frac{15}{7})}{42}\approx 0.74399386$$

Steps Using Definition of Logarithm

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

$$e^{-\frac{42}{41}x}=\frac{7}{15}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-\frac{42}{41}x})=\log(\frac{7}{15})$$

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

$$-\frac{42}{41}x\log(e)=\log(\frac{7}{15})$$

Divide both sides by \log(e).

$$-\frac{42}{41}x=\frac{\log(\frac{7}{15})}{\log(e)}$$

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

$$-\frac{42}{41}x=\log_{e}\left(\frac{7}{15}\right)$$

Divide both sides of the equation by -\frac{42}{41}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{7}{15})}{-\frac{42}{41}}$$

Solve for x (complex solution)

$$x=-\frac{i\times 41\pi n_{1}}{21}+\frac{41\ln(\frac{15}{7})}{42}$$

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