How to Take the Natural Log of a Fraction With X in the Denominator

By Petra Wakefield

One way to find the natural logarithm of a fraction is to first convert the fraction to decimal form, then take the natural log. If the fraction includes a variable, however, this method won't work. When you come across the natural log of a fraction with x in the denominator, turn to the properties of logarithms to simplify the expression. Use the property related to division: log(x/y) = log(x) - log(y).

Rewrite the natural log of the fraction as the natural log of the numerator minus the natural log of the denominator. If your problem is ln(5/x), for example, rewrite it as ln(5) - ln(x).

Take the natural log of the numerator using a scientific calculator. For example, ln(5) = 1.61.

Record the answer using your calculated value. For example, ln(5/x) = 1.61 - ln(x).

Tip

If your natural log is part of an algebraic equation, solve the equation using the value of the natural log. For example, if you have the equation 5 = ln(5/x), plug in 1.61 - ln(x): 5 = 1.61 - ln(x). Rearrange the equation to get ln(x) = -3.39. Raise e to the power of both sides: e^[ln(x)] = e^3.39. Raising e to the power of ln(x) results in x, so x = e^3.39 = 29.7.

About the Author

Petra Wakefield is a writing professional whose work appears on various websites, focusing primarily on topics about science, fitness and outdoor activities. She holds a Master of Science in agricultural engineering from Texas A&M University.