Last updated on July 7th, 2025
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.
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.
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:
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.
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.
Take a look at the graphs of exponential and logarithmic functions for a better understanding.
The key properties of logarithmic graphs are listed below:
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.
The key properties of logarithmic functions are useful when working with exponents and solving equations involving logarithms.
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 integral formulas of logarithmic functions are:
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.
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.
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:
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.
Express 5^4 = 625 in logarithmic form.
log5 (625) = 4
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.
Solve log2 (x) = 4.
16
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.
Solve the expression: log5 (x) = 2.
25
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.
Convert log3 (81) into base 10 (common logarithm).
4
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.
Solve log2 (x) = 5
32
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.
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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