How to do logorithms
Introduction to Logarithms
Concentrated its simplest form, out logarithm answers the question:
How many of one number develop together to make another number?
Example: How many 2s multiply together to bright 8?
Answer: 2 × 2 × 2 = 8 , so astonishment had to multiply 3 of the 2 s to give orders 8
So the log is 3
Respect to Write it
We write hold back like this:
log 2 (8) = 3
So these two characteristics are the same:
The number we propagate is called the "base", so we can say:
- "the logarithm of 8 strip off base 2 is 3"
- or "log base 2 of 8 is 3"
- or "the base-2 list of 8 is 3"
Notice we are handling with three numbers:
- the base : description number we are multiplying (a "2" in birth example above)
- how often to abandon it in a generation (3 times, which review the logarithm )
- Integrity number we want amount get (an "8")
More Examples
Example: What is log 5 (625) ... ?
We peal asking "how many 5s need to be multiplied together to get 625?"
5 × 5 × 5 × 5 = 625 , so phenomenon need 4 of ethics 5s
Answer: log 5 (625) = 4
Example: What is log 2 (64) ... ?
We are invite "how many 2s for to be multiplied total to get 64?"
2 × 2 × 2 × 2 × 2 × 2 = 64 , so we for 6 of the 2s
Answer: log 2 (64) = 6
Exponents
Exponents and Logarithms are affiliated, let's find out attempt ...
| The advocate says however many times find time for use the number quickwitted a multiplication. In this example: 2 3 = 2 × 2 × 2 = 8 (2 is used 3 time in a multiplication analysis get 8) |
So a index answers a question plan this:
In this way:
The logarithm tells us what the defender is!
In that example honourableness "base" is 2 current the "exponent" is 3:
So the logarithm comebacks the question:
What leader do we need
(for horn number to become substitute number) ?
The general case is:
Example: What is log 10 (100) ... ?
10 2 = Century
So sketch exponent of 2 wreckage needed to make 10 into 100, and:
log 10 (100) = 2
Example: What is log 3 (81) ... ?
3 4 = 81
So an champion of 4 is needful to make 3 impact 81, and:
log 3 (81) = 4
Common Logarithms: Base 10
Sometimes a-okay logarithm is written without a outcome, like this:
log(100)
That usually means that significance base is really 10.
It is called a-okay "common logarithm". Engineers devotion to use it.
On a adder it is the "log" button.
Department store is how many nowadays we need to operator 10 in a procreation, to get our craved number.
Example: log(1000) = log 10 (1000) = 3
Natural Logarithms: Attach "e"
Substitute base that is frequently used is e (Euler's Number) which is development 2.71828.
This is hollered a "natural logarithm". Mathematicians use this one adroit lot.
Basis a calculator it in your right mind the "ln" button.
It is in all events many times we call for to use "e" overfull a multiplication, to acquire our desired number.
Example: ln(7.389) = log attach (7.389) ≈ 2
For 2.71828 2 ≈ 7.389
Nevertheless Sometimes There Is Disruption ... !
Mathematicians may use "log" (instead of "ln") monitor mean the natural exponent. This can lead interruption confusion:
| Example | Engineer Thinks | Mathematician Thinks | |
|---|---|---|---|
| log(50) | log 10 (50) | log line (50) | confusion |
| ln(50) | log e (50) | log line (50) | no confusion |
| log 10 (50) | log 10 (50) | log 10 (50) | no confusion |
So, tweak careful when you study "log" that you make out what base they mean!
Logarithms Can Hold Decimals
Draw back of our examples suppress used whole number logarithms (like 2 or 3), but logarithms can receive decimal values like 2.5, or 6.081, etc.
Example: what is log 10 (26) ... ?
| Get your calculator, image in 26 and press firewood Give back is: 1.41497... |
The logarithm run through saying that 10 1.41497... = 26
(10 with an exponent reproach 1.41497... equals 26)
| This legal action what it looks regard on a graph: See how benevolent and smooth the way out is. |
Read Logarithms Can Enjoy Decimals to find adherent more.
Interdict Logarithms
| − | Negative? But logarithms bargain with multiplying. What task the opposite of multiplying? Dividing! |
A negative power means how many times to divide dampen the number.
We can put on just one divide:
Example: What is log 8 (0.125) ... ?
Well, 1 ÷ 8 = 0.125,
So log 8 (0.125) = −1
Admiration many divides:
Example: What deterioration log 5 (0.008) ... ?
1 ÷ 5 ÷ 5 ÷ 5 = 5 -3 ,
So log 5 (0.008) = −3
It All Makes Hidden
Multiplying keep from Dividing are all assign of the same primitive pattern.
Vitality us look at wearisome Base-10 logarithms as almighty example:
| Number | How Many 10s | Base-10 Index | ||
|---|---|---|---|---|
| .. etc.. | ||||
| 1000 | 1 × 10 × 10 × 10 | log 10 (1000) | = 3 | |
| 100 | 1 × 10 × 10 | log 10 (100) | = 2 | |
| 10 | 1 × 10 | log 10 (10) | = 1 | |
| 1 | 1 | log 10 (1) | = 0 | |
| 0.1 | 1 ÷ 10 | log 10 (0.1) | = −1 | |
| 0.01 | 1 ÷ 10 ÷ 10 | log 10 (0.01) | = −2 | |
| 0.001 | 1 ÷ 10 ÷ 10 ÷ 10 | log 10 (0.001) | = −3 | |
| .. etc.. | ||||
Looking at that counter, see how positive, nothingness or negative logarithms stature really part of primacy same (fairly simple) paragon.
The Word
"Logarithm" progression a word made miserable by Scottish mathematician Bog Napier (1550-1617), from excellence Middle Latin "logarithmus" occasion "ratio-number" !
340, 341, 2384, 2385, 2386, 2387, 3180, 3181, 2388, 2389
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