What is logarithm of a number? Explain with one example.
Answers
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In its simplest form, a logarithm answers the question:
How many of one number do we multiply to get another number?
Example: How many 2s do we multiply to get 8?
Answer: 2 × 2 × 2 = 8, so we had to multiply 3 of the 2s to get 8
So the logarithm is 3
How to Write it
We write "the number of 2s we need to multiply to get 8 is 3" as:
log2(8) = 3
So these two things are the same:
logarithm concept 2x2x2=8 same as log_2(8)=3
The number we multiply is called the "base", so we can say:
"the logarithm of 8 with base 2 is 3"
or "log base 2 of 8 is 3"
or "the base-2 log of 8 is 3"
Notice we are dealing with three numbers:
the base: the number we are multiplying (a "2" in the example above)
how often to use it in a multiplication (3 times, which is the logarithm)
The number we want to get (an "8")
More Examples
Example: What is log5(625) ... ?
We are asking "how many 5s need to be multiplied together to get 625?"
5 × 5 × 5 × 5 = 625, so we need 4 of the 5s
Answer: log5(625) = 4
Example: What is log2(64) ... ?
We are asking "how many 2s need to be multiplied together to get 64?"
2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s
Answer: log2(64) = 6
Exponents
Exponents and Logarithms are related, let's find out how ...
2 cubed
The exponent says how many times to use the number in a multiplication.
In this example: 23 = 2 × 2 × 2 = 8
(2 is used 3 times in a multiplication to get 8)
So a logarithm answers a question like this:
2 with what exponent = 8
In this way:
2^3=8 becomes log_2(8)=3
The logarithm tells us what the exponent is!
In that example the "base" is 2 and the "exponent" is 3:
2^3=8 becomes log_2(8)=3
So the logarithm answers the question:
What exponent do we need
(for one number to become another number) ?
The general case is:
a^x=y becomes log_a(y)=x
Example: What is log10(100) ... ?
102 = 100
So an exponent of 2 is needed to make 10 into 100, and:
log10(100) = 2
Example: What is log3(81) ... ?
34 = 81
So an exponent of 4 is needed to make 3 into 81, and:
log3(81) = 4