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 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

$y = l n x$ 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 1:

l n 1 = 0

because e⁰ = 1

Property 2:

l n e = 1

because e¹ = e

Property 3: If

\ln x = \ln y

, then x = y One-to One Property

Property 4:

l n e^{x} = x

, and

e^{\ln x} = x

Inverse Property

Let's solve some simple natural logarithmic equations:

\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 $l n e^{x} = x$ , therefore the left-hand side becomes -1.

$\ln e^{- 1} = x$ Rewrite

-1 = x Apply Property

Example 1:

l n x = l n 3 x - 28

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 $\ln x = \ln y$ , 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:

\frac{l n 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.

$\frac{0}{20} = x + 3$ Apply Property

Step 3: Solve for x.

0 = x + 3 Evaluate LHS

x = -3 Subtract 3

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