Logarithmic Equations: Introduction and Simple Equations

A logarithmic function is the inverse to an 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 common logarithmic functions.

The general common logarithmic equation is:

COMMON LOGARITHMIC FUNCTION

$y = l o g_{a} x$ if and only if x = a^y

Where a > 0, a ≠ 1 and x > 0

When reading

l o g_{a} x

say, "log base a of x".

Some examples are:

1.

l o g_{10} 100 = 2

because 10² = 100

2.

l o g_{3} 81 = 4

because 3⁴ = 81

3.

l o g_{15} 225 = 2

because 15² = 225

Notice in the examples that the base of the log is also the base of the corresponding exponent. In example 1 above, the logarithmic function has a log of base 10 and the corresponding exponential function has a base of 10.

If you see log with no base is means log of base 10 or log = log₁₀.

Some basic properties of logarithmic functions are:

Property 1:

l o g_{a} 1 = 0

because a⁰ = 1

Property 2:

l o g_{a} a = 1

because a¹ = a

Property 3: If

l o g_{a} x = l o g_{a} y

, then x = y One-to One Property

Property 4:

l o g_{a} a^{x} = x

and

a^{{log}_{a} x} = x

Inverse Property

Let's solve some simple logarithmic equations:

log x = 4

Step 1: Choose the most appropriate property.

Properties 1 and 2 do not apply, as the log 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.

Remember $l o g = l o g_{10}$ . Since the log has a base of 10, taking the inverse means to rewrite both sides as exponents with base 10.

log x = 4 Original

10^logx = 10⁴ Exponent of 10

Step 3: Solve for x.

Property 4 states that $a^{l o g_{a} x} = x$ , therefore the left-hand side becomes x.

x = 10⁴ Apply Property

x = 10,000 Evaluate

Example 1:

l o g_{3} x = l o g_{3} 4 x - 9

Step 1: Choose the most appropriate property.

Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Since a log is set equal to a log of the same base. Property 3 is the most appropriate.

Property 3 - One to One

Step 2: Apply the Property.

Property 3 states that if $l o g_{a} x = l o g_{a} y$ , then x = y. Therefore x = 4x - 9.

x = 4x - 9 Apply Property

Step 3: Solve for x.

-3x = -9 Subtract 4x

x = 3 Divide by -3

Example 2:

l o g_{3} 3 x = 5

Step 1: Choose the most appropriate property.

Properties 1 and 2 do not apply, as the log 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.

Since the log has a base of 3, taking the inverse means to rewrite both sides as exponents with base 3.

$l o g_{3} 3 x = 5$ Original

$3^{\log_{3} 3 x} = 3^{5}$ Exponent of 3

Step 3: Solve for x.

Property 4 states that $a^{l o g_{a} x} = x$ , therefore the left-hand side becomes x.

3^x = 3⁵ Apply Property

$x = \frac{243}{3}$ Divide by 3

x = 81 Evaluate

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