Key takeaway: The natural logarithm ln x is the logarithm with base e ≈ 2.71828, so ln x = y means e^y = x, and it undoes the exponential function e^x.
Grade 11 – Algebra 2F-LE.A.4Functions and Graphs
Definition of Natural Logarithm
The natural logarithm is the logarithm whose base is Euler’s number \(e\). It is written \(\ln x\) and answers the question: to what power must \(e\) be raised to give \(x\)?
Because the exponential \(e^x\) models continuous growth and decay, its inverse is the standard way to find a time or a rate. If a quantity grows by the factor \(e^{rt}\), taking \(\ln\) of both sides isolates the exponent \(rt\). Its graph rises slowly, crosses the x-axis at 1, and never touches the y-axis.
Formula and notation
The symbol \(\ln\) stands for “logarithm natural” and replaces \(\log_e\):
\[\ln x = y \iff e^{y}=x, \quad x>0.\]
The two cancellation rules are \(\ln(e^x)=x\) for all real \(x\) and \(e^{\ln x}=x\) for \(x>0\).
Key facts
- ln 1 = 0 and ln e = 1, the two anchor values of the graph.
- The domain is x > 0; the range is every real number.
- ln x is negative for 0 < x < 1 and positive for x > 1.
- A calculator’s LN key gives ln x directly; LOG gives base 10.
Natural Logarithm example
A bacteria sample follows \(N=5e^{0.4t}\), in thousands, with \(t\) in hours. When does it reach 45 thousand?
-
Set \(5e^{0.4t}=45\) and divide by 5: \(e^{0.4t}=9\).
-
Take the natural logarithm of both sides: \(0.4t=\ln 9\).
-
Since \(\ln 9\approx 2.1972\), we get \(t=\frac{2.1972}{0.4}\approx 5.49\).
The sample reaches 45 thousand after about 5.49 hours.
Common mistake
Watch out
Priscilla writes \(\ln(2+3)=\ln 2+\ln 3\approx 0.693+1.099=1.792\).
Why it is wrong
The natural logarithm turns products into sums, not sums into sums. \(\ln(2+3)=\ln 5\approx 1.609\), while \(\ln 2+\ln 3=\ln 6\approx 1.792\). Those are different numbers.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Functions and Graphs