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Introduction and Simple Equations with the Natural Base

For simple equations and basic properties of the natural exponential function see EXPONENTIAL EQUATIONS: Introduction & Simple Equations.

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

Quick Review

The exponential function has the form:

EXPONENTIAL FUNCTION

y = abx

Where a ≠ 0, b ≠ 1 and x is any real number.



The basic properties for the exponential function are:

Property 1: b0 = 1

Property 2: b1 = b

Property 3: bx = by if and only if x = y     One-to One Property

Property 4: logb bx = 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.

12(3x) = 156

Step 1: Isolate the exponent.


In this case divide both sides of the equation by 12.

3x = 13 Divide by 12

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


Since the x is an exponent of base 3, take log3 of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

log 3 3 x = log 3  13   Take log3

Step 3: Apply the Property and solve for x.


Property 4 states lo g b b x =x . Thus the left-hand side becomes x.


To get a value for log3 13 you may need to change to log of base 10. This is covered as a separate topic.


In short, take the log of base 10 of 13 and divided by the log of base 10 of 3, the original base.


lo g 3 13= lo g 10  13 lo g 10  3 = log 13 log 3

x = log3 13 Apply Property


x = log3 13 Exact answer


x= log 13 log3  Change base


x≈2.335 Approximation

Example 1:      6(2(3x+1)) - 8 = 52

Step 1: Isolate the exponent.


In this case add 8 to both sides of the equation. Then divide both sides by 6.

6(2(3x+1)) - 8 = 52 Original


6(2(3x+1)) = 60   Add 8


2(3x+1) = 10     Divide by 6

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


Since the x is an exponent of base 2, take log2 of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

lo g 2   2 3x+1 =lo g 2  10 Take log2

Step 3: Apply the Property and solve for x.


Property 4 states lo g b b x =x . Thus the left-hand side becomes the exponent, 3x + 1. Now isolate the x.


To get a value for log2 10 you may need to change to log of base 10. This is covered as a separate topic.


In short take the log of base 10 of 10 and divided by the log of base 10 of 2, the original base.


lo g 2 10= lo g 10  10 lo g 10  2 = log 10 log 2

3x + 1 = log2 10   Apply Property


3x = log2 10 - 1    Subtract 1


x= lo g 2  10 3 − 1 3     Divide by 3


x= lo g 2  10 3 − 1 3    Exact answer


x= 1 3  ·  log10 log2 − 1 3 Change base


x≈0.774 Approximation

Example 1:      9-3-x = 729

Step 1: Isolate the exponent.


In this case the exponent is isolated.

9-3-x = 729 Original

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


Since the x is an exponent of base 9, take log9 of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

log9 9-3-x = log9 729 Take log9

Step 3: Apply the Property and solve for x.


Property 4 states lo g b b x =x . Thus the left-hand side becomes -3 - x. Now isolate the x.


To get a value for log9 729 you may need to change to log of base 10. This is covered as a separate topic.


In short take the log of base 10 of 729 and divided by the log of base 10 of 9, the original base.


lo g 9 729= lo g 10  729 lo g 10  9 = log 729 log 9

-3 - x = log9 729   Apply Property


-x = log9 729 + 3   Add 3


x = -(log9 729 + 3) Divide by -1


x = -(log9 729 + 3) Exact answer


x=−( log 729 log9 +3 ) Change base


x = 6   Exact value





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