Natural Logarithm and the Number e

In mathematics, some numbers are so important they get their own letter. One of the most famous is Euler's number, denoted by $e$.

The Number $e$

The number $e$ is an irrational mathematical constant, approximately equal to $2.71828$. It naturally arises in problems involving continuous growth, such as calculating compound interest when the number of compounding periods grows without bound.

The exponential function $f(x) = e^x$ is unique because its rate of growth is exactly equal to its current value at any point.

The Natural Logarithm

The natural logarithm is simply a logarithm with base $e$. Instead of writing $\log_e(x)$, we use the special notation $\ln(x)$.

$\ln(x) = \log_e(x)$

Because exponential and logarithmic functions are inverses of each other, $e^x$ and $\ln(x)$ perfectly "undo" one another. This gives us two crucial inverse properties:

1. $e^{\ln(x)} = x$ (for $x > 0$)
2. $\ln(e^x) = x$ (for any real number $x$)

Additionally, because $\ln$ is just a base-$e$ logarithm, it shares all the standard logarithm properties, plus a few easy-to-remember facts:

• $\ln(e) = 1$
• $\ln(1) = 0$

Solving Equations with $e$ and $\ln$

Understanding the inverse relationship makes solving equations straightforward. Let's look at a couple of examples.

Example 1: Simplify $e^{\ln 5}$

Using the inverse property $e^{\ln(x)} = x$, the $e$ and the $\ln$ cancel each other out.

$e^{\ln 5} = 5$

Example 2: Solve $e^{2x} = 7$

To get the variable $x$ out of the exponent, we can take the natural logarithm of both sides of the equation:

$\ln(e^{2x}) = \ln(7)$

Using the property $\ln(e^y) = y$, the left side simplifies to $2x$:

$2x = \ln(7)$

Now, just divide by $2$ to isolate $x$:

$x = \frac{\ln(7)}{2}$

(Note: You can leave the answer in this exact form, or use a calculator to find the approximate decimal value, which is about $0.973$.)