# Exponential Equations - Complex Equations

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. 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. x = log3 13 Apply Property x = log3 13 Exact answer  Change base $x\approx 2.335$ 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. 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. 3x + 1 = log2 10   Apply Property 3x = log2 10 - 1    Subtract 1     Divide by 3    Exact answer Change base $x\approx 0.774$ 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. -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 Change base x = 6   Exact value

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

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