Natural Base - Simple Equations
This discussion will focus on the natural logarithmic functions.
A natural log is a log with base e. The base e is an irrational number, like π, that is approximately 2.718281828.
Instead of writing loge, the natural logarithm has its own symbol, ln. In other words, loge x = ln x
The general natural logarithmic equation is:
NATURAL LOGARITHMIC FUNCTION
if and only if x = ey
Where a > 0
When reading ln x say, "the natural log of x".
Some basic properties of natural logarithmic functions are:
Property 2: because e1 = e
Property 3: If , then x = y One-to One Property
Property 4: , and Inverse Property
Let's solve some simple natural logarithmic equations:
Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the ln equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate. |
Property 4 - Inverse |
Step 2: Apply the Property. First rewrite as an exponent. Property 4 states that , therefore the left-hand side becomes -1. |
Rewrite -1 = x Apply Property |
Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the ln equals neither 0 nor 1. Since a natural log is set equal to another natural log, Property 3 is the most appropriate. |
Property 3 - One to One |
Step 2: Apply the Property. Property 3 states that if , then x = y. Therefore x = 3x - 28. |
x = 3x - 28 Apply Property |
Step 3: Solve for x. |
-2x = -28 Subtract 3x x = 14 Divide by -2 |
Step 1: Choose the most appropriate property. Property 1 applies as it states that ln 1 = 0. |
Property 1 |
Step 2: Apply the Property. Rewrite the left-hand side replacing ln 1 with 0. |
Apply Property |
Step 3: Solve for x. |
0 = x + 3 Evaluate LHS x = -3 Subtract 3 |
Related Links: Math algebra General Differentiation Rules Completing the Square when a = 1 Completing the Square when a ≠ 1 Algebra Topics |
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