Natural Logarithm

High School

Definition

A logarithm to the special base of base e (e = 2.7182818...). You may see it being written as ln(x) or logₑ(x). Natural logarithm functions are the inverse functions of exponential functions, which allows you to rewrite a natural logarithm in exponential form by converting back and forth.

Worked examples

\(\ln(e) = 1 \quad \ln(e^3) = 3 \quad \ln(1) = 0\)

The natural log asks 'e to what power gives this?' so \(\ln(e^x) = x\).

\(\ln(20) \approx 2.996\)

Most natural logs are irrational; use a calculator or leave in log form.

\(\ln(x) = 4 \)→\( e^4 = x \)→\( x \approx 54.598\)

Convert to exponential form to solve: if \(\ln(x) = y\) then \(e^y = x\).

Common mistakes

\(\ln(a + b) = \ln(a) + \ln(b)\) → \(\ln(a \cdot b) = \ln(a) + \ln(b)\) Logs turn multiplication into addition, not addition into addition.
\(\ln(e^2) = e^2\) → \(\ln(e^2) = 2\) The natural log and \(e^x\) are inverses, so they cancel: \(\ln(e^x) = x\).
\(\ln(-5)\) is defined → \(\ln(x)\) only exists for \(x > 0\) You cannot take the natural log of zero or a negative number in the real numbers.

Where you'll use it next

You'll use natural logs to solve exponential growth and decay problems in calculus, differentiate and integrate \(e^x\) and \(\ln(x)\), and model real phenomena like population growth, radioactive decay, and compound interest.

Found in 1 StudyPug lesson

Natural log: ln

Algebra 2

11th Grade11th