Logarithmic Functions

In Mathematics, before the discovery of calculus, many Math scholars used logarithms to change multiplication and division problems into addition and subtraction problems. In Logarithms, the power is raised to some numbers (usually, base number) to get some other number. It is an inverse function of exponential function. We know that Mathematics and Science constantly deal with the large powers of numbers, logarithms are most important and useful. In this article, we are going to discuss the definition and formula for the logarithmic function, rules and properties, examples in detail.

Logarithmic Function Definition

In mathematics, the logarithmic function is an inverse function to exponentiation. The logarithmic function is defined as

For x > 0 , a > 0, and a ≠1,

y= log_a x if and only if x = a^y

Then the function is given by

f(x) = log_a x

The base of the logarithm is a. This can be read it as log base a of x. The most 2 common bases used in logarithmic functions are base 10 and base e.

Also, try out: Logarithm Calculator

Common Logarithmic Function

The logarithmic function with base 10 is called the common logarithmic function and it is denoted by log₁₀ or simply log.

f(x) = log₁₀ x

Natural Logarithmic Function

The logarithmic function to the base e is called the natural logarithmic function and it is denoted by log_e^.

f(x) = log_e x

Logarithmic Functions Properties

Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below.

Product Rule

log_b MN = log_b M + log_b N

Multiply two numbers with the same base, then add the exponents.

Example : log 30 + log 2 = log 60
Quotient Rule 

log_b M/N = log_b M – log_b N

Divide two numbers with the same base, subtract the exponents.

Example : log₈ 56 – log₈ 7 = log₈(56/7)=log₈8 = 1

Power Rule 

Raise an exponential expression to power and multiply the exponents.

Log_b M^p = P log_b M

Example : log 100³ = 3. Log 100 = 3 x 2 = 6

Zero Exponent Rule

log_a 1 = 0.

Change of Base Rule

log_b (x) = ln x / ln b or log_b (x) = log₁₀ x / log₁₀ b

Other Important Rules of Logarithmic Function

• Log_b b = 1 Example : log₁₀10 = 1
• Log_b b^x = x Example : log₁₀10^x = x
• \(\begin{array}{l}b^{\log _{b}x}=x. \text{ Substitute } y= log_{b}x \text{, it becomes } b^{y} = x\end{array} \)

There are also some of the logarithmic function with fractions. It has a useful property to find the log of a fraction by applying the identities

• ln(ab)= ln(a)+ln(b)
• ln(a^x) = x ln (a)

We also can have logarithmic function with fractional base.

Consider an example,

By the definition, log_a b = y becomes a^y = b

(4/9)^y = 27/8

(2²/3²)^y = 3³ / 2³

(⅔)^2y = (3/2)³

Video Lesson

Logarithmic Equations

Logarithmic Function Examples

Here you are provided with some logarithmic functions example.

Example 1:

Use the properties of logarithms to write as a single logarithm for the given equation: 5 log₉ x + 7 log₉ y – 3 log₉ z

Solution:

By using the power rule , Log_b M^p = P log_b M, we can write the given equation as

5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ x⁵ + log₉ y⁷ – log₉ z³

From product rule, log_b MN = log_b M + log_b N

5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ x⁵y⁷ – log₉ z³

From Quotient rule, log_b M/N = log_b M – log_b N

5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ (x⁵y⁷ / z³ )

Therefore, the single logarithm is 5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ (x⁵y⁷ / z³ )

Question 2:

Use the properties of logarithms to write as a single logarithm for the given equation: 1/2 log₂ x – 8 log₂ y – 5 log₂ z

Solution:

By using the power rule , Log_b M^p = P log_b M, we can write the given equation as

1/2 log₂ x – 8 log₂ y – 5 log₂ z = log₂ x^1/2 – log₂ y⁸ – log₂ z⁵

From product rule, log_b MN = log_b M + log_b N

Take minus ‘- ‘ as common

1/2 log₂ x – 8 log₂ y – 5 log₂ z = log₂ x^1/2 – log₂ y⁸z⁵

From Quotient rule, log_b M/N = log_b M – log_b N

1/2 log₂ x – 8 log₂ y – 5 log₂ z = log₂ (x^1/2 / y⁸z⁵ )

The solution is

\(\begin{array}{l}\frac{1}{2} log_{2} x – 8 log_{2} y – 5 log_{2} z = \log _{2}\left ( \frac{\sqrt{x}}{y^{8}z^{5}} \right )\end{array} \)

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