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### Course: Algebra (all content) > Unit 11

Lesson 22: Introduction to logarithms (Algebra 2 level)- Intro to logarithms
- Intro to Logarithms
- Evaluate logarithms
- Evaluating logarithms (advanced)
- Evaluate logarithms (advanced)
- Relationship between exponentials & logarithms
- Relationship between exponentials & logarithms: graphs
- Relationship between exponentials & logarithms: tables
- Relationship between exponentials & logarithms

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# Evaluating logarithms (advanced)

Sal evaluates log₂(8), log₈(2), log₂(⅛), and log₈(½). Created by Sal Khan.

## Want to join the conversation?

- I felt my brain straining at time stamp3:18. It seems in these problems that you just have to recognize roots to solve these logs without a calculator or one is doomed to trial and error. How does one solve LOG problems, without calculator or prefigured tables, on numbers that really don't come out nicely like LOG base 3 of 13?(111 votes)
- One doesn't, really. This describes two methods to calculate logs by hand; one is limitedly useful and the other requires calculating square roots (which can also be done by hand).

http://fiziko.bureau42.com/teaching_tidbits/manual_logarithms.pdf

http://fiziko.bureau42.com/teaching_tidbits/manual_square_root.pdf(64 votes)

- At the first you've said that log 2 to the base 8 will be 1/3 and log 1/8 to the base 2 will be -3,but that's confusing why can't log 2 to base 8 be -3?(24 votes)
- This might help. A power of:

3 means that we multiply a base by itself 3 times.

-3 means that we divide 1 by the base 3 times

1/3 means that we are looking for the cube root of the base.

In general terms a certain power......

P means that we multiply a base by itself P times.

-P means that we divide 1 by the base P times

1/P means that we are looking for the P root of the base.

Think about it, and notice that......

log(base 8)2=1/3 → is the logarithmic form

base 8^(1/3)=2 → is the exponential form

2=cube root of 8

and

log(base 2)1/8=-3 → is the logarithmic form

base 2^-3=1/8 → is the exponential form

1/8=1/(2*2*2) or 1/2/2/2

Also always keep in mind that exponents are practically the opposite of logarithms.

So just remember that.......

log(B)R=P → is the logarithmic form

B^P=R → is the exponential form

where:

B=base (the starting number, and if it is just log R then and there is no base B specified then that means our base B is an automatic 10)

R=resulting number after raising the base B by a certain power of P.

P=power(92 votes)

- I still do not understand how can exponents be fraction...(21 votes)
- Well, think about when you learned multiplication. At first you were told that multiplication is just adding a number to itself the indicated number of times. But, then you learned about multiplying by fractions and decimals -- but how could add an number to itself 2.13 times?

Likewise, with exponents, they tell you that it is just multiplying a number by itself that number of times. But, now you're learning, just like with multiplication, the exponents aren't so limited -- they can be any number at all, just like you can multiply by any number at all.

So, now you are just moving beyond where exponents are so simply described and getting into where they are really useful. In real-world, applied mathematics (for example, things like finance and economics) your exponents are rarely integers.(36 votes)

- In this video, Sal gave us a problem. He said that 8^x=2. so, how did he get 1/3 as a value for x?(12 votes)
- 8^x = 2

if you break the 8 into (2*2*2)^x = 2 it´s the same as (2^3)^x = 2, you know that tou can take an expoent inside parenthesis and multiply by the outside expoent, thus:

(2)^3*x = 2 or 2^1.

same bases? cancel the 2´s, now you have 3x = 1, x = 1/3(14 votes)

- What does 0^0 evaluate to? Because 0 raised to anything is zero, But anything raised to zero is 1... I'm really confused... Is it a complex number by any chance?

thanks!(17 votes)- Don't listen to the last two comments. 0^0 is undefined because there are two conflicting ways to find a solution to this problem.(9 votes)

- How do I find a logarithm of a prime number?(12 votes)
- Did you mean by:

"prime# as for the exponent part"?

"prime# as for the base"?

or " prime # as for the solution set"?

Well, by reasonable sense, I will assume you mean the 3rd condition, which is definitely possible to be written as logarithm. e.g.:

log 9 (1/2)= 3

or

log 125 (1/3) = 5

I hope that's the answer you've been looking for :-)(8 votes)

- Can somebody please explain what Sal said starting from0:40? It confuses me.(4 votes)
- First: Are you aware that a root, such as a square root or a cube root, is the same thing as a fractional exponent? Specifically,

√x = x^(½)

∛x = x^(⅓)

∜x = x^(¼)

And so on...

Thus log₈ (2) = x, can be rearranged to 8ˣ = 2

So, to find what x is, you have to know what power you have to put 8 to to get 2. Since 2³ = 8, then ∛8 = 2

But ∛8 is the same thing as 8^(⅓).

So, the power you have to raise 8 to in order to get 2 is ⅓. Thus, x=⅓ whether you write the problem in the log form or in the exponential form.(19 votes)

- Ok, I understand that exponents can be fractions and I understand how Sal evaluated it. So would log(1023) be 3.009? I'm not allowed to use a calculator.(6 votes)
- To understand the reason why log(1023) equals approximately 3.0099 we have to look at how logarithms work. Saying log(1023) = 3.009 means 10 to the power of 3.009 equals 1023. The ten is known as the base of the logarithm, and when there is no base, the default is 10. 10^3 equals 1000, so it makes sense that to get 1023 you have to put 10 to the base of a number close to 3. So while you might not be able to figure out exactly what log(1023) is without a calculator, you are able to get a good idea of what it should be.(13 votes)

- I think Sal has made a mistake, 8^1/4 is 2 not 8^1/3. 8^1/3 is 2 2/3 which is roughly 2.66666... recurring.(0 votes)
- Incorrect. There is a difference between ^ and *. You are thinking of 8*1/4, which is 2.

The ^ indicates the exponent. This uses multiplication, but is not multiplication. Normally you're used to seeing ^2, or squared. This would mean you repeat the number 2 times and then multiply, so 8*8 = 64.

Does this only works when you see the whole numbers? NO!

The fraction 1/2 is the inverse of 2 (or 2/1). Therefore the operation is the inverse, so instead of squared, we need square root. For example 9^1/2 = 3.

Now for the 1/3. If it was 3/1, you would say cubed. Therefore since we have the inverse fraction, we instead need the cubed root. This means what number multiplied by itself 3 times results in the number 8?

This answer is 2. 2*2*2=4*2=8. Therefore 8^1/3=2.(34 votes)

- Explain how you get 10^0=1(4 votes)
- Anything to the zeroth power is equal to 1. This can be proven in multiple ways.

But let us look at an informal proof:

10⁴/10 = 10³ ←**notice that I subtracted 1 from the exponent**

10³/10 = 10² ←**again we subtracted 1 from the exponent**

10²/10 = 10¹ ←**again we subtracted 1 from the exponent**

10¹/10 = 10⁰ ←**again we subtracted 1 from the exponent**

But 10¹/10 is just 10/10, which is obviously equal to 1

Thus, 10⁰ = 1

So, any number to the zeroth power is just another way of saying a number divided by itself, which is always equal to 1 with the single exception of 0/0 (thus 0⁰ is a special case).

What 0⁰ equals depends on whom you ask. To date, there has been no rigorous mathematical proof that establishes what it equals. However, there are rather a lot of mathematical theorems that suggest but don't prove that 0⁰ = 1, thus most mathematicians would state that 0⁰ = 1. However, not all agree, some claim 0⁰ is undefined or indeterminate.(14 votes)

## Video transcript

Let's give ourselves a
little bit more practice with logarithms. So just as a little bit of
review, let's evaluate log base 2 of 8. What does this evaluate to? Well, it's asking us
or it will evaluate to the power or
the exponent 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. And I'll color-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 1/3. 8 to the one-third
power is equal to 2, or you could say the
cube root of 8 is 2. So in this case, x is 1/3. This logarithm right over
here will evaluate to 1/3. Fascinating. Let's mix it up a
little bit more. Let's say we had the log base 2. Instead of 8, let's put
a 1/8 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 1/8. So if we said that
this is equal to x, we're essentially saying 2 to
the x power is equal to 1/8. Well, we know 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 1/8,
which is the reciprocal of 8, we just have to raise 2
to the negative 3 power. 2 to the negative
3 power is 1 over 2 to the third power, which
is the same thing as 1/8. So if we're asking
ourselves, what exponent do we have to raise
2 to to get to 1/8? 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 1/2? What does this evaluate to? Let me clean this up so that 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 1/2? 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 1
over 8 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 1/2. So the log base 8 of 1/2 is
equal to-- well, what power do I have to raise 8 to to
get to 1/2-- is negative 1/3. Equal to negative 1/3. I hope you enjoyed
that as much as I did.