Natural Exponential Equations - Complex Equations
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 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.
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 Approximation |
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 Divide by 2 Exact answer Approximation |
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 Divide by -7 Exact answer Approximation |
Related Links: Math algebra Exponential Equations - Complex Equations Exponential Equations: Compound Interest Application Algebra Topics Exponential Functions |
To link to this Natural Exponential Equations - Complex Equations page, copy the following code to your site: