To learn about logarithmic functions, first, recall exponential functions because logarithmic functions are the inverses of the exponential functions. The general form of the exponential function is y = a^x, where a represents the base and x represents the power.

The inverse of y = a^x is x = a^y. The expression x = a^y. is equivalent to  y = log_a(x), where a > 0 and a \neq 1. In words, we can describe it as the logarithm of x to the base a is y. Keep in mind that it is not possible to take the logarithm of a negative number or 0.

For example, consider the simple exponential function 9 ^ 2 = 81. In words, we can say that nine raised to the power 2 is 81. Now, we will see how to write this function in logarithmic form and say in words.

In the logarithmic form, we can write this expression as log_9 81 = 2. In words, we can describe it as the logarithm of 81 to the base 9 is 2. In the following table, we have written some common functions in exponential and logarithmic forms.

Exponential formLogarithmic form

From the above table, we can conclude that the base of exponential and its corresponding logarithmic function is the same.

Natural and common logarithms

Natural logarithms are logarithmic functions with base e. Natural log is expressed as

ln(x) = log_{e}(x)

Common logarithms are the logarithmic functions with base ten. They are expressed as follows:

log x = log _{10} x

Before proceeding to the graphs of logarithmic functions, we will evaluate some logarithmic functions first in the following examples.

Example 1

log_{14} 196 = ?

Solution

We will solve the above expression in multiple steps.

Step 1 – Convert in exponential notation

Suppose log_{14} 196 = x. First, express this function in exponential notation:

14^x=196

Step 2 – Solve for x

The exponent we require on 14 to get 196 as an answer is 2, hence log_{14} 196 = 2.

Example 2

log_6 \frac{1}{216} = ?

Solution

Step 1 – Convert in exponential notation

Suppose log_6 \frac{1}{216} = x.

Convert the above function in exponential form:

6^x = \frac {1}{216}

Step 2 – Solve for x

We know that when an exponent is negative, then final answer is a fraction. In the above example, the exponent must be negative because the answer \frac {1}{216} is in fractional form. If the exponent of 6 is 3, we get the answer 216. If the exponent is -3, we will get \frac{1}{216}.

 

Example 3

log _ \frac{2}{7} \frac {4}{49} = ?

Solution

Step 1 – Convert in exponential notation

Suppose log _ \frac{2}{7} \frac {4}{49} = x. First, convert the function in exponential form like this:

\frac {2}{7} ^ x = \frac{4}{49}

Step 2 – Solve for x

Since, \frac {2^2}{7^2} = \frac {4}{49}, so log _ \frac{2}{7} \frac {4}{49} = 2.

 

Example 4

log _ \frac{5}{3} \frac {125}{27} = ?

Solution

Step 1 – Convert in exponential notation

Suppose log _ \frac{5}{3} \frac {125}{27} = x. Convert this logarithmic function into an exponential form like this:

\frac {5}{3} ^ x = \frac{125}{27}

Step 2 – Solve for x

Since, \frac {5^3}{3^3} = \frac {125}{27}, so log _ \frac{5}{3} \frac {125}{27} = 3

 

Characteristics of Logarithmic Functions Graphs

We know how to evaluate the logarithmic functions. Now, we will see how to plot logarithmic functions in a coordinate axis. But before proceeding to some examples, let us see what are some of the characteristics of graphs of the logarithmic functions. The graphs of all logarithmic functions will have the following characteristics:

  • The graph of logarithmic functions passes through the points (1,0).
  • If the base of a logarithmic function is greater than 1, then the graph increases
  • If the base of the logarithmic functions is greater than 0 but smaller than 1, then the graph decreases.
  • The domain of the logarithmic function is all numbers greater than 0.
  • The range of logarithmic function is all real numbers
  • Graph of these functions is asymptotic to the y-axis, i.e. it gets closer to the y-axis but never crosses it.

 

Example 1

Graph the function f(x) = 3 ^ x and is logarithmic form log_3 (x).

Solution

Step 1

We will construct two separate tables for both the functions. The table of values for the exponential function f(x) = 3 ^ x is given below:

xf(x)
-21/9
-11/3
01
13
29

Step 2

Plotting the above values in a coordinate axis will give the following graph of the exponential function:

Example 1- Exponential function graph

Step 3

We know that the logarithmic functions are the inverses of exponential functions. So, the function y = 3 ^ x will be written as log_3 f(x) = x in logarithmic form. The table of this function is given below:

xf(x)
1/9-2
1/3-1
10
31
92

Step 4

The graph of the logarithmic function is given below. You can see that this logarithmic graph is the flipped version of the exponential function graph.

Example 1 - Logarithmic function graph

You can see that the above logarithmic graph satisfies all the properties of the logarithmic function graph. The above logarithmic curve passes through the points (1,0). Although it is near to the y-axis, however, it does not cross it. Since, the base of this logarithmic function was 3, i.e. greater than 1, so the graph is increasing.

 

Example 2

Now, let see in another example what happens when the base of the logarithmic function is less than 1. Plot the graph of the function y = \frac{1}{3} ^ (x) and its logarithmic form log_\frac{1}{3}(x).

Solution

Step 1

To plot the graphs, first we need to construct the table of exponential function like this:

xf(x)
-29
-13
01
11/3
21/9

Step 2

The graph of this function is given below.

Example 2 - Exponential function graph

Step 3

The table of values for the logarithmic function is given below.

xf(x)
9-2
3-1
10
1/31
1/92

Step 4

The graph of the logarithmic function is given below.

Example 2 - Logarithmic function graph

Like all logarithmic functions, the above logarithmic curve passes through the points (1,0). It is also asymptotic to the y-axis, i.e. it gets very close to it, but never crosses it. From the above two examples we can conclude that whether the base of the logarithmic functions is greater than or less than 1, the general properties of logarithmic function graphs do not change. You can see that the graph is decreasing because the value of the base was less than 1.

 

Example 3

Graph the function y = 5 ^ (x) and its logarithmic form log_5(x).

Solution

Step 1

Construct a table of the function y = 5 ^ (x).

xf(x)
-21/25
-11/5
01
15
225

 

Step 2

Plot the graph of the exponential function using the above table of values.

Example 3 - Exponential function graph

Step 3

We know that the logarithmic function is inverse of an exponential function, so we will reverse the values of x and y of the exponential table.

xf(x)
1/25-2
1/5-1
10
51
252

Step 4

We will get the following graph by plotting the values in the x-y plane.

Example 3 - Logarithmic function graph

The above logarithmic curve passes through the points (1,0). It is also asymptotic to the y-axis, i.e. it gets very close to it, but never crosses it. Since, the base of this logarithmic function was 5, i.e. greater than 1, so the graph is increasing.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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