# Logarithmic Equations: Natural Base - Simple Equations

A natural logarithmic function is the inverse to a natural exponential function. Just like exponential function have common bases and a natural base; logarithmic functions have common logs and a natural log.

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

$y=ln{x}_{}$     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 1: because e0 = 1

Property 2: because e1 = e

Property 3: If $\mathrm{ln}x=\mathrm{ln}y$, then x = y     One-to One Property

Property 4: , and ${e}^{\mathrm{ln}x}=x$     Inverse Property

Let's solve some simple natural logarithmic equations:

$\mathrm{ln}\frac{1}{e}=x$
 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 , therefore the left-hand side becomes -1. $\mathrm{ln}{e}^{-1}=x$ Rewrite -1 = x     Apply Property
Example 1:
 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
Example 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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