# Exponential Equations: Simple Equations with the Natural Base

An exponential function has the form y = abx where the base b > 1 and x is any real number.

In many situations the base e is used. The base e is called the natural base and is an irrational number that is approximately 2.718281828.

The natural exponential function has the form:

NATURAL EXPONENTIAL FUNCTION

y = aex

Where a ≠ 0.

Some examples are:

1. y = ex (Where a = 1)

2. y = 65ex (Where a = 65)

3. y = -3ex (Where a = -3)

The properties for the natural base are:

Property 1: e0 = 1

Property 2: e1 = e

Property 3: ex = ey if and only if x = y     One-to One Property

Property 4: ln ex = x           Inverse Property

Just as logarithms are inverse functions to exponents, the inverse function to ex is ln x, called the natural log. This is shown in Property 4.

Let's solve some simple natural exponential equations:

ex = e12
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since both terms are natural exponents, Property 3 is the most appropriate. Property 3 - One to One Step 2: Apply the Property. The equation is already written in the form of bx = by ex = e12 Step 3: Solve for x. Property 3 states ex = ey if and only if x = y, therefore x -12. x = 12
Example 2:      ex = 41
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since 41 can't accurately be written as an exponent with base e, the most appropriate property is the Inverse property, Property 4 Property 4 - Inverse Step 2: Apply the Property To apply Property 4, take the ln of both sides of the equation. ln ex = ln 41 Step 3: Solve for x. Property 4 states that ln ex = x, therefore the left-hand side becomes x. x = ln 41

 Related Links: Math algebra Natural Exponential Equations - Complex Equations Exponential Equations - Complex Equations Algebra Topics Exponential Functions

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