DOMAIN AND RANGE OF LOGARITHMIC FUNCTIONS

About the topic "Domain and range of logarithmic functions"

"Domain and range of logarithmic functions" is a much needed stuff required by almost all the students who study math in high schools.

Even though students can get this stuff on internet, they do not understand exactly what has been explained. 

To make the students to understand the stuff "Domain & range of logarithmic functions", we have given a table which clearly says the domain and range of different logarithmic functions.

Names of the parts of a logarithm

Before knowing domain and range of logarithmic functions, we have to be aware of the names of the parts a logarithm.

Usually a logarithm consists of three parts.

Let us come to the names of those three parts with an example.

log₀ (x) = y               

In the above logarithmic function, 

"10" is called as "Base"

"x" is called as"Argument"

"y" is called as "Answer"

In the topic "Domain and range of logarithmic functions", next we are going to see the fact which is about the domain of logarithmic functions.

Fact to be known about domain of logarithm functions

A very important fact that we have to know about the domain of a logarithm to any base is,

"A logarithmic function is defined only for positive values of argument"

For example, if the logarithmic function is y = log₁₀ (x), then the domain is x>0. 

In the logarithmic function y = log₁₀ (x), argument is "x". Since logarithmic function is defined for only positive values of argument, "x" has to be positive. 

So the domain of the logarithmic function y = log₁₀ (x)  is "x>0"

In the topic "Domain and range of logarithmic functions", next we are going to see the domain of different logarithmic functions.

Domain of y = log₁₀ (x) 

In the logarithmic function y = log₁₀ (x), the argument is "x"

From the fact explained above, argument must always be a positive value.

So, the values of x must be greater than zero. 

So, the domain of the logarithmic function y = log₁₀ (x) is "x > 0"

Domain of y = log₁₀ (x+a)    

In the logarithmic function y = log₁₀ (x+a), the argument is "x+a"

From the fact explained above, argument must always be a positive value.

So, (x+a) > 0 =====> x > -a

So, the domain of the logarithmic function y = log₁₀ (x+a) is "x > -a"

Domain of y = log₁₀ (x-a) 

In the logarithmic function y = log₁₀ (x-a), the argument is "x-a"

From the fact explained above, argument must always be a positive value.

So, (x-a) > 0 =====> x > a

So, the domain of the logarithmic function y = log₁₀ (x+a) is "x > a"

Domain of y = log₁₀ (kx)

In the logarithmic function y = log₁₀ (kx), the argument is "kx"

From the fact explained above, argument must always be a positive value.

So, (kx) > 0 =====> x > 0

So, the domain of the logarithmic function y = log₁₀ (kx) is "x > 0"

Domain of y = log₁₀ (kx+a)

In the logarithmic function y = log₁₀ (kx+a), the argument is "kx+a"

From the fact explained above, argument must always be a positive value.

So, (kx+a) > 0 =====> kx > -a ====> x > -a/k

So, domain of the logarithmic function y = log₁₀ (kx+a) is "x > -a/k"

Domain of y = log₁₀ (kx-a)

In the logarithmic function y = log₁₀ (kx-a), the argument is "kx-a"

From the fact explained above, argument must always be a positive value.

So, (kx-a) > 0 =====> kx > a ====> x > a/k

So, domain of the logarithmic function y = log₁₀ (kx-a) is "x > a/k"

In the topic "Domain and range of logarithmic functions", next we are going to see some more stuff on domain of different logarithmic functions.

Some more stuff on domain of logarithmic functions

Let us consider the logarithmic functions which are explained above.

y = log₁₀ (x)

y = log₁₀ (x+a)

y = log₁₀ (x-a)

y = log₁₀ (kx)

y = log₁₀ (kx+a)

y = log₁₀ (kx-a)

Domain is already explained for all the above logarithmic functions with the base "10".

In case, the base is not "10" for the above logarithmic functions, domain will remain unchanged. 

For example,

in the logarithmic function y = log₁₀ (x), instead of base "10", if there is some other base, again the domain is same. That is "x > 0"

In the topic "Domain and range of logarithmic functions", next we are going to see the range of logarithmic functions.

Range of logarithmic functions

The table given below clearly explains how the range of y = log₁₀ (x) is "All Real Numbers". 

Here, we may think that if the base is not 10, what could be the range of the logarithmic functions?

Whatever base we have for the logarithmic function, the range is always "All Real Numbers".

For the base other than "10", we can define the range of a logarithmic function in the same way as explained above for base "10". 

You can also visit the following web pages to know more about "Domain and range of logarithmic functions".

http://www.analyzemath.com

http://www.mathsisfun.com

http://hotmath.com

https://www.ixl.com

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