All About Logarithmic Functions

Logarithmic functions are fundamental concepts in mathematics that were introduced by the Scottish mathematician John Napier in the 17th century. These functions act as a translator, converting the exponential equations that are easier to understand and manipulate. In this article, we will discuss what logarithm functions are, types, rules, properties, and some essential logarithmic functions, and at the end, we will discuss some solved examples to get a better understanding of logarithmic functions.

So, let's get started.

Table of Content

• What is a Logarithm?
• Types of Logarithm
• Rules and Properties of Logarithms
• Industry-Wise Application of Logarithmic Function
• Logarithm Question

What is a Logarithm?

A logarithmic function is an inverse of an exponential function, i.e., the logarithmic of a number x to the base b is the exponent to which b must be raised to produce x.

For Example, the logarithm of 8 to the base 2 is 3 because 2 is raised to the power 3 is 8.

Similarly, the logarithm of 100 to the base 10 is 2 because 10 raised to the power 2 is 100.

In simple terms, a logarithm is the power to which a number (the base) is raised to get another number.

Don't worry if you didn't get what happened in the above example. We have covered all of them in the upcoming sections.

Representation of Logarithmic Function

For any exponential function Y = a^x, the logarithmic function will be log_a(Y) = x. 

Where a > 0 and referred to as the base.

Let's take some examples:

Example: Change the given exponential function (2³ = 8) into the logarithmic function.

Here, 2 is the base, 3 is the exponent, and 8 is the result obtained. This exponent function can also be written as log₂(8) = 3.

Let's take one more example.

Example: Change the given logarithmic function (log₃(9) = 2) into the exponential function.

Here, the base is 3, so the exponent form will be 3² = 9.

Domain and Range of Logarithmic Function

• Domain of y = log (x) is x > 0 or (0, + infinity).
• The range of any log function is a set of all real numbers.

Types of Logarithm

On a broader level, there are two types of logarithmic functions.

Common Logarithmic Function

Logarithmic function with base 10. It is mainly used in science and engineering because they are easy to calculate and understand.

Example: log 10 (10000).

Note: When you are using "log" without the base, it usually refers to the common logarithmic functions. 

Natural Logarithmic Function

The logarithmic function with the base e (an irrational number equals 2.71828). It is denoted by "ln" and is used in calculus, complex analysis, and describing natural phenomena like population growth, interest calculation, and radioactive decay.

Example: ln (10)

Apart from these common and natural logarithmic functions, there are two other types of logarithmic functions which are often used.

Binary Logarithmic

Logarithmic function with base 2. It measures the power to which the number 2 must be raised to obtain a specific value. Binary logarithmic functions are the backbone of computing, from calculating the complexity of algorithms to determining the storage capacity in bits and bytes.

Example: log 2 (24)

Logarithmic to base n

These are the custom-tailored logarithmic functions, where 'n' can be any positive real number.

Example: log 5 (10) 

Rules and Properties of Logarithmic

Product Rule

log_b​(m × n) = log_b​(m) + log_b​(n)

$Examples$

• log⁡(2×3) = log⁡(2) + log⁡(3)
• log⁡₅(5×25) = log⁡₅(5) + log⁡₅(25)
• ln⁡(e × e²) = ln⁡(e)+ln⁡(e²)

Quotient Rule

log_b​(m/n​) = log_b​(m)−log_b​(n)

$Examples$

• log⁡(8/2) = log⁡(8) − log⁡(2)
• log⁡₃(27/3) = log⁡₃(27) − log⁡₃(3)
• ln⁡(1/e) = ln⁡(1) − ln⁡(e)

Power Rule

log_b​(mⁿ) = n⋅log_b​(m)

$Examples$

• log⁡(10⁴) = 4⋅log⁡(10)]
• log⁡₂(8²) = 2⋅log⁡₂(8)
• ln⁡(e³) = 3⋅ln⁡(e)

Change of Base Rule

log_b​(m) = log_k​(m) / log_k​(b)

Examples​

• log⁡₂(8) = log⁡(8) / log⁡(2)​
• log⁡₅(125) = ln⁡(125) / ln⁡(5)​
• log⁡₇(49) = log⁡₁₀(49) / log⁡₁₀(7)

Logarithm of 1

log_b​(1) = 0

$Examples$

• log⁡(1) = 0
• log⁡₅(1) = 0
• ln⁡(1) = 0

Logarithm of Base

log_b​(b) = 1

1. log⁡(10) = 1
2. log⁡2(2) = 1
3. $\ln (e)$

Industry-Wise Application of Logarithmic Function

Logarithmic Questions for Practice

• Solve for x in the equation log⁡₂(x) + log⁡₂(x−3) = 3.
• If log⁡(a) = 2 and log⁡(b) = 3, find the value of log⁡(a²b)/log(a^(1/2)​).
• Prove that log⁡₁₀(2) is irrational.
• Simplify log⁡₅(125) + 1/2log⁡₅(16)−log⁡₅(8).
• If $x$, express log⁡₃(81) in terms of $x$.
• Prove that for any positive a, $b$, and $c$, the inequality log_a​(bc) > log_b​(ac) + log_c​(ab) holds if and only if a, $b$, and $c$ are the sides of an acute triangle.
• Solve the logarithmic equation for $x$: 2^{log⁡₂(x − 1)} = x² − 4x + 4.
• Show that log_x​(y)⋅log_y​(x) = 1 for all positive $x$ and $y$, x ≠ 1, y ≠ 1.
• If log⁡10(x) + log⁡10(x − 1) = 1, find the value of $x$.
• Given that $a$ and $b$, express log⁡₁₀(1.2) in terms of $a$ and $b$.
• If $a$ and $=b$, find log⁡₁₂(216) in terms of $a$ and $b$.
• Determine the sum S=log⁡₂(3) + log⁡₄(3) + log⁡₈(3) + … + log_2ⁿ​(3).
• Prove that log₂​(x) ⋅ log₃​(x) ⋅ log₄​(x) is constant for all $x$ in the domain of the function.
• Solve for $x$ in the equation log_x+1​(2x²)=2.
• If log_a​(bc), log_b​(ac), and log_c​(ab) are in arithmetic progression, prove that $a$, $b$, and $c$ are in geometric progression.