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Logarithm basics
Fancier logarithm expressions Examples of logarithms that evaluate to fractional and negative values.
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- Let's give ourselves a little more practice
- with logarithms.
- So just a little bit of review...
- Let's evaluate log, base 2, of 8.
- What does this evaluate to?
- Well it's asking us it'll evaluate to the power that I have
- to raise our base to. That I have to raise 2 to,
- to get to 8.
- So 2 to the first power is 2, 2 to the second power is 4,
- 2 to the third power is 8. So this right over here is going to
- be equal to 3.
- Fair enough, we did examples like that in the last video.
- Let's do something a little bit more interesting...
- What is--
- and I'll colour code it...
- what is log, base 8, of 2?
- Now this is interesting, I'll give you a few seconds to
- think about it.
- Well we're asking ourselves or this will evaluate to
- the exponent that I have to raise 8 to, to get to 2.
- So let's think about that in another way.
- So we could say 8, to some power, and that exponent that I'm
- raising 8 to is essentially what this logarithm would evaluate to.
- 8 to some power, is going to be equal to 2.
- Well if 2 to the third power is 8, 8 to the one-third power
- is equal to 2.
- So 'x' is equal to one-third.
- 8 to the one-third power is equal to 2.
- Or you could say that the cube root of 8 is 2.
- So in this case 'x' is one-third.
- This logarithm right over here will evaluate to one-third.
- Fascinating.
- Let's mix it up a little bit more.
- Let's say we have the log, base 2, instead of 8
- let's put a one-eighth, right over here.
- So I'll give you a few seconds to think about that.
- Well it's asking us or this will evaluate to the exponent that I have to
- raise 2 to, to get to one-eighth.
- So if we set this to be equal to 'x', we're essentially saying
- 2 to the 'x' power, is equal to one-eighth.
- Well we know that 2 to the third power--
- Let me write this down...
- We already know that 2 to the third power is equal to 8.
- If we want to get to one-eighth, which is a reciprocal of 8.
- We just have to raise to the negative 3 power.
- 2 to the negative 3 power is one over 2 to the third power.
- Which is the same thing as one over eight.
- So, if we're asking ourselves "what exponent do we have to raise
- 2 to to get to one-eighth?"
- Well we have to raise it to the negative 3 power.
- So 'x' is equal to negative 3.
- This logarithm evaluates to negative 3.
- Now let's really, really mix it up.
- What would be the log, base 8, of one-half.
- What does this evaluate to?
- Let me clean this up so we have some space to work with.
- So as always, we're saying "what power do I have to raise 8 to, to get to one-half?"
- So let's think about that a little bit.
- We already know that 8 to the one-third power is
- equal to 2.
- If we want the reciprocal of 2 right over here, we have to just
- raise 8 to the negative one-third.
- So let me write that down, 8, to the negative one-third power,
- is going to be equal to one over eight, to the one-third power,
- and we already know the cube root of 8 or 8 to the one-third power is
- equal to 2. This is equal to
- one-half.
- So, the log, base 8, of one-half is equal to?
- Well the power I have to raise 8 to to get to one-half is
- negative one-third.
- I hope you enjoyed that as much as I did.
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