Natural Exponential Equations - Complex Equations

For simple equations and basic properties of the natural exponential function see EXPONENTIAL EQUATIONS: Simple Equations With the Natural Base.

This discussion will focus on solving more complex problems involving the natural base. Below is a quick review of natural exponential functions.

Quick Review



The natural exponential function has the form:

NATURAL EXPONENTIAL FUNCTION

y = aex

Where a ≠ 0



The natural base e is an irrational number, like π, which has an approximate value of 2.718.

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


Let's solve some complex natural exponential equations.

Remember when solving for x, regardless of the function type, the goal is to isolate the x-variable.

ex -12 = 47

Step 1: Isolate the natural base exponent.


In this case add 12 to both sides of the equation.

ex = 59

Step 2: Select the appropriate property to isolate the x-variable.


Since the x is an exponent of natural base e, take the natural log of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

ln ex = in 59

Step 3: Apply the Property and solve for x.


Property 4 states ln ex = x. Thus the left-hand side becomes x.

x = ln 59      Apply Property


x = ln 59      Exact answer


x4.078      Approximation

Example 1:      3e2x-5 + 11 = 56

Step 1: Isolate the natural base exponent.


In this case subtract 11 from both sides of the equation. Then divide both sides by 3.

3e2x-5 + 11 = 56   Original


3e2x-5 = 45 Subtract 11


e2x-5 = 15   Divide by 3

Step 2: Select the appropriate property to isolate the x-variable.


Since the x is an exponent of natural base e, take the natural log of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

ln e2x-5 = ln 15 Take ln

Step 3: Apply the Property and solve for x.


Property 4 states that ln ex = x. Thus the left-hand side simplifies to the exponent, 2x - 5.


Next isolate the x but adding 5 and dividing by 2.

2x - 5 = ln 15 Apply Property


2x = ln 15 + 5 Add 5


x= ln15+5 2  Divide by 2


x= ln15+5 2 Exact answer


x3.854  Approximation

Example 2:      1500e-7x = 300

Step 1: Isolate the natural base exponent.


In this case divide both sides of the equation by 1500

1500e-7x = 300     Original


e-7x = 0.2   Divide by 1500

Step 2: Select the appropriate property to isolate the x-variable.


Since the x is an exponent of natural base e, take the natural log of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

ln e-7x = ln 0.2 Take ln

Step 3: Apply the Property and solve for x.


Property 4 states that ln ex = x.


Thus the left-hand side simplifies to the exponent, -7x.


Next isolate the x but dividing by -7.

-7x = ln 0.2 Apply Property


x= ln0.2 7 Divide by -7


x= ln0.2 7  Exact answer


x0.230 Approximation





Related Links:
Math
algebra
Exponential Equations - Complex Equations
Exponential Equations: Compound Interest Application
Algebra Topics
Exponential Functions


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