Solve 1/48=e^-frac{0.035{45}*0.137x}

$$\frac { 1 } { 48 } = e ^ { - \frac { 0.035 } { 45 } \times 0.137 x }$$

Solve for x

$$x=\frac{9000000\ln(48)}{959}\approx 36330.35359559$$

Steps Using Definition of Logarithm

Expand \frac{0.035}{45} by multiplying both numerator and the denominator by 1000.

$$\frac{1}{48}=e^{\left(-\frac{35}{45000}\right)\times 0.137x}$$

Reduce the fraction \frac{35}{45000} to lowest terms by extracting and canceling out 5.

$$\frac{1}{48}=e^{-\frac{7}{9000}\times 0.137x}$$

Multiply -\frac{7}{9000} and 0.137 to get -\frac{959}{9000000}.

$$\frac{1}{48}=e^{-\frac{959}{9000000}x}$$

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

$$e^{-\frac{959}{9000000}x}=\frac{1}{48}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-\frac{959}{9000000}x})=\log(\frac{1}{48})$$

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

$$-\frac{959}{9000000}x\log(e)=\log(\frac{1}{48})$$

Divide both sides by \log(e).

$$-\frac{959}{9000000}x=\frac{\log(\frac{1}{48})}{\log(e)}$$

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

$$-\frac{959}{9000000}x=\log_{e}\left(\frac{1}{48}\right)$$

Divide both sides of the equation by -\frac{959}{9000000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{\ln(48)}{-\frac{959}{9000000}}$$

Solve for x (complex solution)

$$x=-\frac{i\times 18000000\pi n_{1}}{959}+\frac{9000000\ln(48)}{959}$$

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