Logarithmic Functions
Logarithm Properties in Action

We have seen in our section on Logarithms, that Logarithms are a Code for Exponents. The above slide is a review of this concept

When we look an any number raised to a power, that is any number with an exponent, we can think of the exponent as a logarithm or a log as represented by the log icon. Thus when

2³ = 8 can be expressed as log₂ 8 = 3

or we can say that the 'log with base 2 of 8' = 3.

For a power of 10 then,

10³ = 1,000 can be expressed as 3 = log₁₀ 1,000 = 3.

Logarithm Property #1

The first Logarithm Property that we use when working with Logarithmic Functions states that the Logarithm to any base a of (1) is 0.

log_a(1) = 0

We can see that this is true when we use the Logarithm Code and set

x = log_a(1)

This leads us to

a^x=1

Any number raised to the 0 power is 1 so we see that x=0 and

log_a(1) = a⁰ = 1

This can be demonstrated more clearly with a log₁₀ example. We set

x = log₁₀(1)

This leads us to

10^x=10⁰

and

x = 0

so

log₁₀(1) = 0

Logarithm Property #2

The second Logarithm Property that we use to solve and work with Logarithmic Functions states that the Logarithm to any base a of (a) is 1.

log_a(a) = 1

We can see that this is true when we use the Logarithm Code and set

x = log_a(a)

This leads us to

a^x=a¹=a

Any number raised to the power of one is equal to the number itself and thus

log_a(a) = 1

This can be demonstrated more clearly with a log₁₀ example. We set

x = log₁₀(10)

This leads us to

10^x=10 = 10¹

and

x = 1

so

log₁₀(10) = 1

Logarithm Property #3

The third Logarithm Property that we use when working with Logarithmic Functions states that the Logarithm to any base a of a^x is x.

log_a(a^x) = x

We can see that this is true when we use the Logarithm Code and set

x = log_a(a^x)

This leads us to

a^x = a^x

and

x = x

so

log_a(a^x) = x

We can see this more clearly with a log₁₀ example. We set

x = log₁₀(100) = log₁₀(10²)

and

10^x = 10² = 100

so

log₁₀(100) = log₁₀(10²) = 2

Logarithmic Functions - Logarithm Property #4

The fourth Logarithm Property is a very powerful technique for simplifying and solving Logarithmic functions. The fourth Logarithm Property states that "a" raised to the power of [logarithm with base "a" of any number y]is equal to that number y.

This is represented as

a^log_a(y) = y

The best way to see that this Logarithm Property is true is to let

a = 10 and y = 100

This yields

10^{[log₁₀(100)]} = 100

We use the Logarithm Code for log₁₀(100) = x as follows

10^x = 100 = 10²

and therefore

10^x = 10²

and [log₁₀(100)] = 2

so

10^{[log₁₀(100)]} = 10^[2] = 100

Logarithmic Functions - Summary of Logarithm Properties

The four Logarithm Properties are summarized in the above chart. Examples of how these four Logarithm Properties are used to solve Logarithmic Functions are given below.

Example of Logarithm Property #1 - Solve log₅(1) = ?

Using Logarithm Property #1: log_a(1) = 0 we see that

log₅(1) = 0

Example of Logarithm Property #2 - Solve log₈(8) = ?

Using Logarithm Property #2: log_a(a) = 1

log₈(8) = 1

Example of Logarithm Property #3 - Solve log₅(625) = ?

Using Logarithm Property #3: log_a(a^x) = x

and recognizing that 625 = 5⁴

log₅(625) = log₅(5⁴) = 4

Example of Logarithm Property #4 - Solve 7^[log₇(12)] = ?

Using Logarithm Property #4:

a^[log_a(x)] = x

7^[log₇(12)] = 12

Click here for the Secrets of Logarithms

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