Base and Argument of Logarithm

What are Base and Argument in a Logarithm?

Consider the following expression:
`log_{10}(2)`
In the above expression,
  • The number 10 is called the base of the logarithm
  • The number 2 is called the argument of the logarithm
The above expression is read as
Logarithm of 2 to the base 10.
We can see that
  • The base is always written in subscript (below) just after the word 'log'
  • The argument is the number written in brackets after the base. It is written in line with the word 'log'

What do the base and argument mean in a logarithm?

If we refer to a logarithm table, the value of `log_{10}(2)` is 100. That is,
`log_{10}(2) = 100`
We already know that logarithms are the opposite of exponents. The above expression if converted to exponents is
`10^2 = 100`
In the above expression, we know that 10 is the base raised to the power (exponent) of 2. Thus we call 10 the base of the logarithm `log_{10}(2)`

The number 2 is called the argument because logarithm (to the base 10) is a function to which we pass the value '2' in order to get the result 100.

What if you change the base of the above logarithm?

Suppose we change the base of the above logarithm to 5. What will the result be then?
`log_5(2) = ?`
Let the result be 'x'.
`log_5(2) = x`
Since 5 is the base and 2 is the argument, we will raise 5 to the power of 2 to get 'x'.
`5^2 = x`
Evaluate `5^2 = 5 * 5 = 25`, thus,
`5^2 = 25`
Thus x is 25 and the logarithm's result is 25.
`log_5(2) = 25`
Thus, base of a logarithm is as important as its argument. Changing it changes the meaning and value of the logarithm.

Special Cases

  1. If the base of a logarithm is not specified, we assume it to be 10. Thus, `log(2)` is same as `log_{10}(2)`
  2. If the base of a logarithm is not specified and instead of 'log' there is 'ln' (which means natural logarithm) then the base is the exponential constant, `e`. Thus, `ln(10)` is same as `log_e(10)`.

Solved Examples

Identify the base and argument of the following logarithms:
  1. `log_{10}(5)`
  2. `log_{100}(2)`
  3. `log_x(y)`
  4. `log(10)`
  5. `ln(100)`

Answers:

  1. The number written in subscript after log is the base. Thus, 10 is the base. The argument is the number written in line with log after the base. Thus, 5 is the argument.
  2. 100 is the base and 2 is the argument.
  3. The variable 'x' is written in subscript after log. Thus the base is `x`. `y` is written after the base and it is in line with 'log'
  4. No base is specified. Thus the base is 10.
  5. No base is specified and `ln` is written instead of `log`. Thus, the base is the exponential constant, `e`.

Worksheet

Instructions: 

Find the base of the following logarithms

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`log(3)`
Show Answer
`log_{100}(50)`
Show Answer
`log_e(10)`
Show Answer
`log_{a}b`
Show Answer
`log_{2^3}(8)`
Show Answer
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Instructions:

Find the argument of the following logarithms:

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Question
Answer
`log(e)`
Show Answer
`log(10)`
Show Answer
`log_{10}(5)`
Show Answer
`log_{n}(2)`
Show Answer
`ln(2)`
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