Logarithmic Equations: 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 log_{e}, the natural logarithm has its own symbol, ln. In other words, log_{e} x = ln x
The general natural logarithmic equation is:
NATURAL LOGARITHMIC FUNCTION
$y=ln{x}_{}$ if and only if x = e^{y}
Where a > 0
When reading ln x say, "the natural log of x".
Some basic properties of natural logarithmic functions are:
Property 2: $ln\mathrm{e}=1$ because e^{1} = e
Property 3: If $\mathrm{ln}x=\mathrm{ln}y$, then x = y One-to One Property
Property 4: $ln{e}^{x}=x$, and ${e}^{\mathrm{ln}x}=x$ 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 $\frac{1}{e}$ as an exponent. Property 4 states that $ln{e}^{x}=x$, therefore the left-hand side becomes -1. |
$\mathrm{ln}{e}^{-1}=x$ 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 $\mathrm{ln}x=\mathrm{ln}y$, 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. |
$\frac{0}{20}=x+3$ Apply Property |
Step 3: Solve for x. |
0 = x + 3 Evaluate LHS x = -3 Subtract 3 |
Related Links: Math algebra Logarithmic Equations: Introduction and Simple Equations Algebra Topics Logarithmic Equations |
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