Natural Log

What is the Natural Logarithm?

The natural logarithm, written as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is Euler's number—an irrational mathematical constant approximately equal to \(2.7182818284\).

While common logarithms use base \(10\) (representing how many times you must multiply \(10\) to get a number), the natural logarithm is the inverse function of the natural exponential function:

$$y = \ln(x) \iff e^y = x$$

Because the rate of growth of the exponential function \(e^x\) is equal to its value at any point, the natural logarithm is standard in calculus, continuous growth modeling (such as population growth or radioactive decay), and compound interest calculations.

Properties of the Natural Logarithm

The natural logarithm shares the general properties of all logarithmic functions:

• Product Rule: \(\ln(a \cdot b) = \ln(a) + \ln(b)\)
• Quotient Rule: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\)
• Power Rule: \(\ln(a^n) = n \cdot \ln(a)\)
• Identity: \(\ln(e) = 1\) and \(\ln(1) = 0\)

Step-by-Step Calculation Example

Suppose you want to solve the equation \(e^y = 10\). To find the exponent \(y\), you take the natural logarithm of both sides:

1. Start with the equation: \(e^y = 10\)
2. Take the natural logarithm of both sides: \(\ln(e^y) = \ln(10)\)
3. Simplify the left side using the identity \(\ln(e^y) = y\): $$y = \ln(10)$$
4. Using a calculator to compute the value: $$y \approx 2.3025850930$$
5. Thus, \(e^{2.3025850930} \approx 10\).