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Evaluating logarithms (advanced)

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

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  • male robot donald style avatar for user B. Swedlow
    I felt my brain straining at time stamp . 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?
    (99 votes)
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  • leaf green style avatar for user Jushya Siddhartha
    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?
    (22 votes)
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    • old spice man green style avatar for user Petrie (Peter S. Asiain III)
      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
      (85 votes)
  • aqualine ultimate style avatar for user jason huh
    I still do not understand how can exponents be fraction...
    (17 votes)
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    • piceratops ultimate style avatar for user Just Keith
      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.
      (28 votes)
  • leaf green style avatar for user 1688592
    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?
    (11 votes)
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  • spunky sam blue style avatar for user Vrajesh
    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!
    (16 votes)
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  • hopper cool style avatar for user 15tkostolansky
    How do I find a logarithm of a prime number?
    (12 votes)
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    • mr pink red style avatar for user tokyomilmil
      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 :-)
      (7 votes)
  • hopper jumping style avatar for user VICTOR STUDIOS
    Can somebody please explain what Sal said starting from ? It confuses me.
    (4 votes)
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    • piceratops ultimate style avatar for user Just Keith
      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.
      (17 votes)
  • starky sapling style avatar for user 👑💎 Professor Sykes 💎👑
    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)
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    • primosaur tree style avatar for user Bradley Reynolds
      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.
      (12 votes)
  • blobby green style avatar for user Alex Hope
    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)
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    • piceratops ultimate style avatar for user Megan Morgan
      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)
  • blobby green style avatar for user Michael Nilsson
    Explain how you get 10^0=1
    (4 votes)
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    • piceratops ultimate style avatar for user Just Keith
      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.