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 = ae^x

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: e^x = e^y if and only if x = y One-to One Property

Property 4: ln e^x = 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.

e^x -12 = 47

Step 1: Isolate the natural base exponent.

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

e^x = 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 e^x = in 59

Step 3: Apply the Property and solve for x.

Property 4 states ln e^x = 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: 3e^2x-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.

3e^2x-5 + 11 = 56 Original

3e^2x-5 = 45 Subtract 11

e^2x-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 e^2x-5 = ln 15 Take ln

Step 3: Apply the Property and solve for x.

Property 4 states that ln e^x = 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

$x = \frac{\ln 15 + 5}{2}$ Divide by 2

$x = \frac{\ln 15 + 5}{2}$ 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 e^x = 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

$x = \frac{\ln 0.2}{- 7}$ Divide by -7

$x = \frac{\ln 0.2}{- 7}$ Exact answer

$x \approx 0.230$ Approximation

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