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Logarithm basics
Logarithms Intro to logarithms. Learning that logarithms are just a way of expressing "the exponent that you have to raise a base to to get another number"
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- Let's learn a little bit about the wonderful
- world of logarithms.
- So we already know how to take exponents.
- If I were to say 2 to the fourth power, what does that mean?
- Well that means 2 times 2 times 2 times 2.
- 2 multiplied or repeatedly multiplied 4 times, and so
- this is going to be
- 2 times 2 is 4
- times 2 is 8, times 2 is 16.
- But what if we think about things in another way.
- We know that we get to 16 when we raise 2 to some power
- but we want to know what that power is.
- So for example, let's say that I start with 2, and I say
- I'm raising it to some power, what does that power have to be to get 16?
- Well we just figured that out. 'X' would have to be 4.
- And this is what logarithms are fundamentally about, figuring out what power
- you have to raise to, to get another number.
- Now the way that we would denote this with logarithm notation
- is we would say, log, base--
- actually let me make it a little bit more colourful.
- Log, base 2--
- I'll do this 2 in blue...
- Log, base 2, of 16
- is equal to what, or is equal in this case since we have the 'x' there,
- is equal to 'x'?
- This and this are completely equivalent statements.
- This is saying "hey well if I take 2 to some 'x' power I get 16'."
- This is saying, "what power do I need to raise 2 to, to get 16 and I'm going to set that to be equal to 'x'."
- And you'll say, "well you have to raise it to the fourth power and once again
- 'x' is equal to 4.
- So with that out of the way let's try more examples of evaluating logarithmic expressions.
- Let's say you had...
- log, base 3, of 81. What would this evaluate to?
- Well this is a reminder, this evaluates to the power we have to raise 3 to, to get
- to 81.
- So if you want to, you can set this to be equal to an 'x', and you can
- restate this equation as, 3 to the 'x' power, is equal to 81.
- Why is a logarithm useful? And you'll see that it has very interesting properties
- later on.
- But you didn't necessarily have to use algebra. To do it this way, to say that 'x' is the power
- you raise 3 to to get to 81, you had to use algebra here, while with just
- a straight up logarithmic expression, you didn't really have to use any algebra, we didn't have to
- say that it was equal to 'x', we could just say that this evaluates to
- the power I need to raise 3 to to get to 81.
- The power I need to raise 3 to to get to 81.
- Well what power do you have to raise 3 to to get to 81?
- Well let's experiment a little bit,
- so 3 to the first power is just 3, 3 to the second power is 9,
- 3 to the third power is 27, 3 to the fourth power, 27 times 3 is equal to 81.
- 3 to the fourth power is equal to 81.
- 'X' is equal to four.
- So we could say...
- Log, base 3, of 81, is equal to--
- I'll do this in a different colour.
- Is equal to 4.
- Let's do several more of these examples and I really encourage you to give a shot
- on your own and hopefully you'll get the hang of it.
- So let's try a larger number, let's say we want to take
- log, base 6, of 216. What will this evaluate to?
- Well we're asking ourselves, "what power do we have to raise 6 to, to get 216?"
- 6 to the first power is 6, 6 to the second power is 36,
- 36 times 6 is 216.
- This is equal to 216.
- So this is 6 to the third power is equal to 216.
- So if someone says "what power do I have to raise 6 to--
- this base here,
- to get to 216?"
- Well that's just going to be equal to 3.
- 6 to the third power is equal to 216.
- Let's try another one. Let's say I had, I dunno,
- log, base 2, of 64.
- So what does this evaluate to?
- Well once again we're asking ourselves, "well this will evaluate to the exponent that I have to raise this
- base 2, and you do this as a little subscript right here.
- The exponent that I have to raise 2 to, to get to 64."
- So 2 to the first power is 2, 2 to the second power is 4,
- 8, 16, 32, 64.
- So this right over here is 2 to the sixth power, is equal to 64. So when you evaluate this expression
- you say "what power do I have to raise 2 to, to get to 64?"
- Well I have to raise to the sixth power.
- Let's do a slightly more straightforward one, or maybe this will be less
- straightforward depending on how you view it.
- What is log, base 100, of 1?
- I'll let you think about that for a second.
- 100 is a subscript so it's, log, base 100, of 1.
- That's one way to think about it, I'll put parentheses around the 1.
- What does this evaluate to?
- Well this is asking ourselves, or we would evaluate this as,
- "what power do I have to raise 100 to, to get to 1?"
- So let me write this down as an equation.
- If I set this to be equal to 'x', this is literally saying
- 100, to what power, is equal to 1?
- Well anything that a 0 power is equal to 1.
- So in this case 'x' is equal to 0.
- So log, base 100, of 1, is going to be equal to 0.
- Log base anything of 1, is going to be equal to 0 because anything to the 0
- power and we're not talking about 0 here. Anything that is to the power of 0
- that is not 0, is going to be equal to 1.
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