Natural Logarithm (ln)
In short: The natural logarithm, written ln x, reports the power you must give the constant e to land on x, which makes it the exact reverse of the growth function e^x.
Algebra 2Functions and GraphsF-LE.A.4
What is Natural Logarithm?
The natural logarithm is the everyday log of calculus and science. It uses the special base \(e \approx 2.71828\) so that continuous growth and decay problems come out cleanly.
By definition, \(\ln x = y\) means \(e^y = x\), for \(x > 0\). Two facts follow at once: \(\ln(e^k) = k\) and \(e^{\ln x} = x\). Its curve crosses the axis at \((1, 0)\), goes through \((e, 1)\), climbs slowly, and never touches the line \(x = 0\).
How to spot it
- The calculator key ln is needed to rescue a variable that sits in the exponent of e.
- An equation such as e^{0.045t} = 2.5 must be solved for t.
- A doubling time, half-life or time-to-reach-a-target question involves continuous change.
- A result must be left in exact form, like ln 50, before being approximated; the natural logarithm gives that exact form.
The method
- Isolate the exponential part, so the equation looks like \(e^{(\text{something})} = \text{number}\).
- Apply the natural logarithm to both sides; the exponent comes down because \(\ln(e^k) = k\).
- Solve the remaining equation for the variable.
- Approximate with a calculator and test the value in the original equation.
Natural Logarithm: solved example
Problem
A bacteria culture follows \(N = 120e^{0.045t}\), with \(t\) in hours. When will it reach 300 cells?
Solution
Set \(120e^{0.045t} = 300\) and divide by 120: \(e^{0.045t} = 2.5\). Take the natural log of both sides: \(0.045t = \ln 2.5\).
Since \(\ln 2.5 \approx 0.91629\), \(t \approx \frac{0.91629}{0.045} \approx 20.36\). The culture reaches 300 cells after about 20.4 hours.
Try it
Question 1
Simplify \(\ln(e^7)\).
Show the answer
The two operations cancel, so \(\ln(e^7) = 7\).
Question 2
Solve \(\ln x = 2\).
Show the answer
Rewrite as \(x = e^2 \approx 7.389\).
Natural Logarithm vs. Logarithm
A general log can use any positive base other than 1, such as 2 or 10. The natural one is the single special case whose base is e. The notation \(\log x\) usually means base 10, while \(\ln x\) always means base e, so \(\log 100 = 2\) but \(\ln 100 \approx 4.605\).
Keep going
Lessons
Worksheets and quizzes
- Exponential Functions and the Number e – Worksheet
- Exponential Functions and the Number e – Quiz
- Exponential and Logarithmic Equations – Worksheet
- Exponential and Logarithmic Equations – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs