# Complex Equations with the Natural Base

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 $x\approx 4.078$     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 $x=\frac{\mathrm{ln}15+5}{2}$  Divide by 2 $x=\frac{\mathrm{ln}15+5}{2}$ Exact answer $x\approx 3.854$  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 $x=\frac{\mathrm{ln}0.2}{-7}$ Divide by -7 $x=\frac{\mathrm{ln}0.2}{-7}$ Exact answer $x\approx 0.230$ Approximation

 Related Links: Math algebra Introduction and Simple Equations with the Natural Base Compound Interest Application Algebra Topics

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