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: ln 1=0 because e0 = 1

Property 2: ln e=1 because e1 = e

Property 3: If lnx=lny , then x = y     One-to One Property

Property 4: ln  e x =x , and e lnx =x      Inverse Property


Let's solve some simple natural logarithmic equations:

ln 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 1 e as an exponent.


Property 4 states that ln  e x =x , therefore the left-hand side becomes -1.

ln e 1 =x Rewrite


-1 = x     Apply Property

Example 1:      ln x=ln 3x28

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 lnx=lny , 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:      ln 1 20 =x+3

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.

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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