What is log ????
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Answers
Answer:
log2 64 = 6, as 2^6= 64
it is known as logirithm
Step-by-step explanation:
Introduction to Logarithms
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
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 ...
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:
In this way:
The logarithm tells us what the exponent is!
In that example the "base" is 2 and the "exponent" is 3:
So the logarithm answers the question:
What exponent do we need
(for one number to become another number) ?
The general case is:
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
Common Logarithms: Base 10
Sometimes a logarithm is written without a base, like this:
log(100)
This usually means that the base is really 10.
It is called a "common logarithm". Engineers love to use it.
On a calculator it is the "log" button.
It is how many times we need to use 10 in a multiplication, to get our desired number.
Example: log(1000) = log10(1000) = 3
Natural Logarithms: Base "e"
Another base that is often used is e (Euler's Number) which is about 2.71828.
This is called a "natural logarithm". Mathematicians use this one a lot.
On a calculator it is the "ln" button.
It is how many times we need to use "e" in a multiplication, to get our desired number.
Example: ln(7.389) = loge(7.389) ≈ 2
Because 2.718282 ≈ 7.389
But Sometimes There Is Confusion ... !
Mathematicians use "log" (instead of "ln") to describe natural logarithm.
Logarithms Can Have Decimals
All of our examples have used whole number logarithms (like 2 or 3), but logarithms can have decimal values like 2.5, or 6.081, etc.
Example: what is log10(26) ... ?

Get your calculator, type in 26 and press log
Answer is: 1.41497...
The logarithm is saying that 101.41497... = 26
(10 with an exponent of 1.41497... equals 26)
Read Logarithms Can Have Decimals to find out more.
Negative Logarithms
−Negative? But logarithms deal with multiplying.
What is the opposite of multiplying? Dividing!
A negative logarithm means how many times to divide by the number.
We can have just one divide:
Example: What is log8(0.125) ... ?
Well, 1 ÷ 8 = 0.125,
So log8(0.125) = −1
Or many divides:
Example: What is log5(0.008) ... ?
1 ÷ 5 ÷ 5 ÷ 5 = 5−3,
So log5(0.008) = −3
It All Makes Sense !!!