Natural Logarithm

The natural logarithm (base-e-logarithm) of a positive real number x, represented by lnx or log_ex, is the exponent to which the base ‘e’ (≈ 2.718…, Euler’s number) is raised to obtain ‘x.’ 

Mathematically, ln(x) = log_e(x) = y if and only if e^y = x

It is also written as:

$\ln x={\int }_1^x{\displaystyle \frac{1}{t}}dt$

The number 2.718… is used naturally in math and science calculations, like pie (π) in geometry. 

Graphing

The geometric interpretation of the natural logarithmic function y = lnx is shown below.

Here, the natural logarithmic function has an asymptote at x = 0 and an x-intercept at (1, 0).

Proof

Now, let us verify the natural logarithm formula $\ln x={\int }_1^x{\displaystyle \frac{1}{t}}dt$

According to the fundamental theorem of calculus, if F(x) is an antiderivative of f(x) for the interval [a, b], then ${\int }_a^{b}f\left(x\right)dx=f\left(b\right)-f\left(a\right)$

Now, let us find an antiderivative of $f\left(t\right)={\displaystyle \frac{1}{t}}$

$\int {\displaystyle \frac{1}{t}}dt=\ln \left|t\right|+C$, here c is the integrating constant.

Applying the fundamental theorem of calculus, we get

${\int }_1^x{\displaystyle \frac{1}{t}}dt=\ln \left|x\right|-\ln \left|1\right|$

= $\ln \left|x\right|-0$

= $\ln \left|x\right|$

Thus, ${\int }_1^x{\displaystyle \frac{1}{t}}dt=\ln \left|x\right|$, is proved.

Rules

The natural logarithm follows all the properties of the logarithm. 

Product Rule

ln(xy) = lnx + lny

Example: ln(10) = ln(5) + ln(2)

Quotient Rule

$\ln \left({\displaystyle \frac{x}{y}}\right)=\ln x-\ln y$

Example:  $\ln \left({\displaystyle \frac{42}{5}}\right)=\ln 42-\ln 5$

Power Rule

ln(xⁿ) = n lnx

Example:  ln((5z)⁷) = 7 ln(5z)

Change of Base Rule

$\ln x={\log }_{e}x={\displaystyle \frac{{\log }_{b}x}{{\log }_{b}e}}$

Or, lnx ⋅ log_be = log_bx

Example: $\ln \left(2p^2\right)={\log }_e\left(2p^2\right)={\displaystyle \frac{{\log }_b\left(2p^2\right)}{{\log }_{b}e}}$, for any base b

Identity Rule

ln(e) = 1

Zero Rule

ln(1) = 0

Infinity Rule

ln(∞) = ∞, which means while the argument ‘x’ approaches to infinity, the limit of ln(x) is:

lim ln(x) = ∞, when x → ∞

Derivative Rule

${\displaystyle \frac{d}{dx}}(\ln \left(x\right))={\displaystyle \frac{1}{x}}$

Integration Rule

$\int \ln \left(x\right)dx=x(\ln \left(x\right)-1)+c$

Others

• ln of Negative Numbers: ln of any negative number is undefined.
• ln of ‘0’ and the Limit: ln(0) is undefined, which means while the argument ‘x’ approaches zero, the limit of ln(0) is minus infinity. That is, lim ln(0) = -∞, when x → 0⁺
• ln of ‘e’ raised to the power ‘x’: ln(e^x) = x
• ‘e’ raised to the ln power: e^ln(x) = x

Solved Example