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Last updated on July 7th, 2025

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Logarithmic Functions

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In the 16th century, a Scottish mathematician, astronomer, and scientist named John Napier discovered the logarithmic function. Logarithms are very useful for solving problems in mathematics, scientific calculations, astronomy, and population studies.A logarithmic function is the inverse of an exponential function. The exponential function ax = N can be converted to the log function, loga (N) = x. It indicates the power to which the base (a) must be raised to get a certain result. In this article, we will explore logarithmic functions and their properties in detail.

Logarithmic Functions for US Students
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What are Logarithmic Functions?

In mathematics, a key function that serves as the inverse of an exponential function is the logarithm function. The basic form of a logarithmic function is:   
  

f(x) = loga (x) or y = loga (x)

Here, a > 0 and a ≠ 1. The exponential function of the above log form is:
   ay = x
Natural logarithm (ln) and common logarithm (log) are the two types of logarithmic functions. 
   

For example, f(x) = ln (x - 2) represents a natural logarithmic function, while g(x) = log (x + 5) -2 represents a common logarithmic function. 

 

The logarithmic functions help to solve exponential equations, especially when the exponent is not an integer. For instance, 2x = 10 can be transformed to log2 (10) = x, and we can easily find the value of x, even if it is not a whole number. 
The formula for converting an exponential function into a logarithmic function is:       

 

A logarithm indicates the exponent to which base must be raised to get a value inside the log. Logarithms cannot be used for negative numbers, but they can be used for decimals, fractions, and whole numbers.

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Domain and Range of Log Functions

The logarithm of zero or negative numbers cannot be calculated. Basic logarithmic function is:

f(x) = log (x) (or y = log (x) ) where x>0 


The set of all positive real numbers is known as the domain ( x > 0) or (0, ∞ ). The output value, y, can be any real number, positive, negative, or zero. A list of y-values for various x-values is provided here:

 

  • When x = 1, y = log (1) = 0
     
  • When x = 2, y = log (2) ≈ 0.3010 
     
  • When x = 0.2, y = log (0.2) ≈  - 0.6990 
     
  • When x = 0.01, y = log (0.01) = - 2

 

The range of every logarithmic function is always a real number, and the domain is always greater than 0. For example, the domain and range of the logarithmic function f(x) = log (x + 3).

 

The argument of the function should be greater than 0 to determine the domain. 
Now, we can solve for x:
      x + 3 > 0
      x > -3 
Thus, the domain is x > -3 or (-3, ∞).

Next, we can find the range; it can be any real number. Therefore, the range is Range = R. 

 

Hence, the range of f(x) = R. 

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Logarithmic Graph

A logarithmic graph depicts how the logarithmic function varies with different values of input x. The domain is the set of positive real numbers, while the range is a set of any real numbers. Concerning the line y = x, the logarithmic and exponential function graphs are symmetrical. It means that the graph of an exponential function across the line y = x is reflected in the graph of the logarithmic function. 

 

  • y = 0 when x = 1 because loga (1) 0 for any base a. Then the graph has an x-intercept.  

 

  • loga (0) is not defined, so the graph does not have a y-intercept. 

 

Take a look at the graphs of exponential and logarithmic functions for a better understanding. 

 

The key properties of logarithmic graphs are listed below:

 

  • a > 0 and a ≠ 1. It explains that the base must be greater than 0 and it should not be equal to 1. 

 

  • When a > 1, the logarithmic graph increases, while the graph decreases when 0 < a < 1. 

 

  • The input to the function will always have a positive value. 

 

  • The range or the output of a function can be any real number. 
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Graphing Logarithmic Functions

Finding the points that illustrate the behavior of a function and drawing a curve through them is known as graphing logarithmic functions. Depending on the base value, the direction of the curve either increases or decreases. 

 

The curve increases if the base is larger than 1 (base > 1), and it decreases if the base is between 0 and 1 (0 < base < 1). To graph a logarithmic function, we have to follow certain steps, they are:
 

Step 1: Identify the domain and range. 

 

Step 2: Find the vertical asymptote by setting the argument equal to 0. Remember that logarithmic graphs have a vertical asymptote but no horizontal asymptote.

 

Step 3: Set the argument equal to 1 by substituting a value of x. To find the x-intercept, use the property loga (1) = 0. 

 

Step 4: Set the argument equal to the base by substituting the value of x. To find another point on the graph, use the property loga (a) = 1. 

 

Step 5: Draw a curve by joining the two points and extending the curve towards the vertical asymptote.

 

Let us take a look at this example to make it easier to understand. 
Graph the logarithmic function f(x) = log2 (x - 1)

 

Step 1: The basic form of logarithmic function is f(x) = loga (x). 
Where a is the base, hence, the base is 2, and since 2 > 1, the curve will increase.  

 

Step 2: Now, set the argument greater than 0. 
   x - 1 > 0 
   x > 1 
Thus, domain = (1, ∞ ). 
Range = R

 

Step 3: Find the vertical asymptote by setting the argument equal to 0. In a logarithmic function, the argument must be positive ( > 0). 
     x - 1 = 0 
     x = 1 
Hence, the vertical asymptote is at x = 1. 

 

Step 4: Next, we can find the points. 
At x = 2:
     f(2) = log2 (2 - 1) = log2 (1) = 0 
So the point is (2, 0)

At x = 3: 
     f(3) = log2 (3 - 1) = log2 (2) = 1 
Hence, the point is (3, 1)

 

Step 5: Draw the graph by joining the points, starting from near the asymptote x = 1. 

  

Here, the curve passes through the points (2, 0) and (3, 1), and it increases slowly. The red line indicates the vertical asymptote at x = 1. 

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What are the Properties of Logarithmic Functions?

The key properties of logarithmic functions are useful when working with exponents and solving equations involving logarithms. 

 

  • Multiplication property: logb (a × b) = logb (a) + logb (b) 
    When two numbers are multiplied inside a logarithm, you can separate them into a sum of two logs. 

 

  • Division property: logb (a/b) = logb (a) - logb (b)
    Dividing two numbers inside a logarithm is equivalent to subtracting the logarithm of the denominator from the numerator. 

 

  • Change of base rule: logb (a) = logc (a) / logc (b)
    You can change the base by dividing the logarithm of the number by the logarithm of the new base. 

 

  • Power property: logb (ax) = x logb (a) 
    To remove the exponent inside a logarithm, multiply the logarithm by that exponent. 

 

  • Logarithm of 1: logb (1) = 0 
    Regardless of the base, the logarithm of 1 is always 0. 

 

  • Logarithm of the base: logb (b) = 1
    If the base and number inside the logarithm are the same, the result is always 1.
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Derivative and Integral of Logarithmic Functions

A logarithmic function’s derivative explains how the function varies with changes in its input. The reverse of the derivative is the integral of the logarithmic function, which helps to find the original function from its rate of change. The derivative formula for the common and natural logarithmic functions are:

 

  • The derivative of the natural logarithm (ln x):
    d / dx [ln x] = 1 / x  

 

  • The derivative of logarithm with base a (loga x):
    d / dx (loga x) = 1 / (x ln a)

 

The integral formulas of logarithmic functions are: 

 

  • The integral of the natural logarithm (ln x): 
    ∫ ln x dx = x (ln x - 1) + C 

 

This indicates that when we integrate ln x, we get a formula that includes x, in x, and a constant C. The accumulated area under the curve of ln x is represented by the constant C. 

 

  • The integral of the common logarithm (log x):
    ∫ log10 x dx = x (log10 x - 1) + C 

 

We get a formula that includes x, log x, and a constant C when we integrate log x. The area under the curve of the common logarithm is represented by the constant C.  

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Real-Life Applications of Logarithmic Functions

Learning the concept of logarithmic functions helps us to apply them to various real-life situations. Here are some real-world applications of logarithmic functions: 

 

  • Logarithmic functions are used by seismologists to measure seismic waves and analyze earthquakes. They measure the magnitude of earthquakes using the Richter scale, which represents large numbers using logarithmic functions. 

 

  • In scientific laboratories, scientists use the pH scale, which is based on a logarithmic scale, to measure the acidity of various solutions. For example, a solution with pH 4 is ten times more acidic than a solution with pH 5. 

 

  • Shareholders and investors can use the logarithmic functions to calculate their compound interest and total investment rate. For instance, they can calculate the time it takes for an investment to reach a certain amount with a fixed interest rate.
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Common Mistakes and How to Avoid Them on Logarithmic Functions

Working with logarithmic functions can be challenging, and students often make mistakes when using them. Here are some common mistakes and their helpful solutions to prevent them while solving mathematical problems. 

Mistake 1

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Assuming the Argument is Negative

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Always remember that the argument inside the logarithm is positive. Sometimes, students mistakenly think that the argument is negative, which will lead them to incorrect answers. 

 

For example, logb (-4) is not defined. 
logb (4) is valid and defined.

Mistake 2

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Forgetting that the Domain must be a Positive Real Number 

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Do not forget the condition that the domain of the logarithm must be a positive number. It is the set of positive real numbers, and it must be greater than 0. It can be expressed as:

 ( x > 0) or (0,∞ )

Mistake 3

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Confusion Between Logarithms and Exponents

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Keep in mind that logarithms are the inverse of exponents. For the exponent form by = x, the logarithm is logb (x) = y. Confusion between them will lead students to wrong conclusions. 

 

For instance, a logarithm form is:

 log2 (8) = 3 is correct. 

The exponential form is:

23 = 8

Mistake 4

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Misinterpreting the Multiplication Property

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Students should learn the product rule that can be useful for multiplication inside the logarithm. Remember to add the logarithms only for multiplication; otherwise, they will end up with incorrect values. Multiplying two numbers inside a logarithm is equal to adding the logarithms together.

Mistake 5

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Ignoring the Simplification of Logarithms Before Solving

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Always remember to simplify logarithmic expressions before equations are solved. If students forget to simplify the expression, then it will become complicated to understand. 

 

For example, log (3x) = log (3) + log (x)

This simplifies the equation and makes it easier to understand. 

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Solved Examples of Logarithmic Functions

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Problem 1

Express 5^4 = 625 in logarithmic form.

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log5 (625) = 4

Explanation

The exponential form ax = N can be written in a logarithmic form as loga (N)  = x.

Hence, 54 can be written as log5 (625) = 4.

Thus, the logarithmic form is log5 (625) = 4. 

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Problem 2

Solve log2 (x) = 4.

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16

Explanation

The logarithm form loga (N) = x is the inverse of the exponential function ax = N.

 

The form log2 (x) = 4 means that 24 = 16.

Since 24 = 2 × 2 × 2 × 2 = 16

  x = 16

 

Thus, the value of x is 16.

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Problem 3

Solve the expression: log5 (x) = 2.

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25

Explanation

The given logarithm form is:
     log5 (x) = 2
 

It is the inverse of the exponential form:
      52 = x

 

Now, solve for x:
    x = 5 × 5 = 25
    x = 25 

 

The value of x is 25.

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Problem 4

Convert log3 (81) into base 10 (common logarithm).

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4

Explanation

Here, we can use the formula for the change of base:
     logb (a) = logc (a) / logc (b)

Now, we can substitute the values:
    Log3 (81) = log (81) / log (3) 

To find the value of log (81) and log (3), using a scientific calculator.
      log (81) ≈ 1.9085
      log (3) ≈ 0.4771 

Thus, log3 (81) = 1.9085 / 0.4771 ≈ 4

Therefore, log3 (81) = 4.

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Problem 5

Solve log2 (x) = 5

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32

Explanation

The given logarithmic form is: 
    log2 (x) = 5

 

It is the inverse of the exponential form:
      25 = x 

 

Hence, x = 2 × 2 × 2 × 2 × 2 = 32
     x = 32 

 

Thus, the value of x is 32.

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FAQs on Logarithmic Functions

1.What do you mean by logarithm?

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2.What is the basic form of logarithmic functions?

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3.Explain the product rule.

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4.Differentiate domain and range.

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5.Explain the quotient rule.

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6.How does learning Algebra help students in United States make better decisions in daily life?

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7.How can cultural or local activities in United States support learning Algebra topics such as Logarithmic Functions ?

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8.How do technology and digital tools in United States support learning Algebra and Logarithmic Functions ?

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9.Does learning Algebra support future career opportunities for students in United States?

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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