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Proof of the logarithm change of base rule

Sal proves the logarithmic change of base rule, logₐ(b)=logₓ(b)/logₓ(a). Created by Sal Khan.

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  • leafers seedling style avatar for user gunhoo93
    At , why did he take both sides of equation with log base b, what is the reasoning behind it?
    More precisely how did someone discover that doing above would prove the formula?
    (17 votes)
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    • male robot hal style avatar for user Saavan Kiran
      aaah gunhoo93 I have asked myself this question an uncountable number of times. Each time in school we learn some theorem or law and I see the proof and I can't stop asking myself so how did they figure out to take these steps to prove something they didn't even know they were trying to prove in the first place? After years of thinking about it starting from 6th grade I've realized that Mathematicians have been throwing around equations and manipulating them for thousands of years. So every proof out there has been discovered either by the theory itself being obvious but it hasn't been proven so someone tried to come up with a proof with that as its end goal OR people just manipulating equations and previously proven things into different equations and discovering these kinds of rules and laws. As I said mathematics has been around for thousands of years and so all of mathematics has been made up of the basics or fundamentals being manipulated or used to make other things and those things used to make other things in a trial and error fashion to get where it is now. For this specific proof see how first they used the more basic logarithm and then manipulated it into the fraction? Someone long ago must have been playing around with it and ran into that equation. I know this is a lot to take in. Even me after realizing this am still trying to wrap my head around it but Im only in 8th grade so I may be wrong. But I hope this answered your question even though I answered it 5 years later.
      (22 votes)
  • mr pants teal style avatar for user Excalibur2000
    How can logs be applied in something like word problems? Does anyone have an example?
    (9 votes)
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    • male robot hal style avatar for user Jesse
      The mass of a certain radioactive sample is 3.42 kilograms when discovered. A week later, the mass of the sample is 2.87 kilograms. What is the half-life of the substance?

      This problem requires us to model radioactive decay with an exponential function. Since the half-life is an unknown in the exponent of the model, you must solve for it using a logarithm.
      (25 votes)
  • leaf grey style avatar for user Roger Weong
    So you can introduce a logarithms into an equation and treating it like an algebraic term?
    (7 votes)
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  • aqualine ultimate style avatar for user Simum
    Why can you take logs both sides ? Similarly write both sides as an exponent ? Is both sides equal ❗
    (5 votes)
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    • female robot ada style avatar for user Angelica Chen
      An equal sign is a powerful thing in mathematics. There is almost nothing that if you perform on one entire side that you cannot do on the other. I think this may just be one of the axioms (foundational rules) of algebra. Let me think if there is anything that you cannot do to both sides at once...

      You cannot divide by zero on both sides.
      You cannot multiply, add, etc. infinity to both sides. Or I guess you can, but it would turn your equation into 0=0, so it's useless.

      Well, that's all I can think of, but there may be some more exceptions. Comments are welcome here.
      (10 votes)
  • aqualine tree style avatar for user rose
    Can I prove this just by trying some numbers?
    (3 votes)
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    • leaf green style avatar for user kubleeka
      No. That would only prove it for those numbers. For all you know, you got lucky and picked numbers that happen to work.

      The idea of a proof is to show that something is true beyond any doubt. Not reasonable doubt. Not justified doubt. Any doubt. And unless you can try every real number (you can't), there will still be doubt.
      (12 votes)
  • stelly green style avatar for user Archita
    I think that I saw this in some other questions, but if on my calculator there's a "log" button with no other options to add numbers, should I assume that it has a base 10?
    (4 votes)
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  • duskpin tree style avatar for user m pe
    Does anybody know where I can find pratice questions with answers using the logarithm rule (a being the base & the log being negative" - ") -log a(b)=loga(b^-1)=log a(1/b) the change of base rule is at the end like in exponents a^-b=1/a^b
    I googled but didn't find what I was looking for.
    (3 votes)
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  • hopper cool style avatar for user coco170
    At , why is logarithm base b of a to the y is the same as y times the logarithm base b of a?
    (4 votes)
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  • male robot hal style avatar for user CadeSkywalker
    At of the video, Sal introduces the log_b to both sides. Why is this logarithm needed all of a sudden?
    (4 votes)
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  • blobby green style avatar for user jintian.peng
    Is there a simpler or easier to understand proof of the change of base rule? I'm confused about why and how you add log base 10 to both side.
    (5 votes)
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Video transcript

What I want to do in this video is prove the change of base formula for logarithms, which tells us-- let me write this-- formula. Which tells us that if I want to figure out the logarithm base a of x, that I can figure this out by taking logarithms with a different base. That this would be equal to the logarithm base b-- so some other base-- base b of x, divided by the logarithm base b of a. And this is a really useful result. If your calculator only has natural logarithm or log base 10, you can now use this to figure out the logarithm using any base. If you want to figure out the log base 2-- let me make it clear. If you want to figure out the logarithm base, let's say, base 3 of, let's say, 25, you can use your calculator either using log base 10 or log base 2. So you could say that this is going to be equal to log base 10 of 25-- and most calculators have a button for that-- divided by log base 10 of 3. So this is an application of the change of base formula. But let's actually prove it. So let's say that we want to-- let's set logarithm base a of x to be equal to some new variable. Let's call that variable, let's call that equal to y. So this right over here, we are just setting that equal to y. Well, this is just another way of saying that a to the y power is equal to x. So we can rewrite this as a to the y power is equal to x. I'll write the x out here, because I'm about to-- these two things are equal. This is just another way of restating what we just wrote up here. Now, let's introduce the logarithm base b. And to introduce it, I'm just to take log base b of both sides of this equation. So let's take logarithm base b of the left-hand side, and logarithm base b of the right-hand side. Well, we know from our logarithm properties that the logarithm of something to a power is the exact same thing as the power times the logarithm of that something. So logarithm base b of a to the y is the same thing as y times the logarithm base b of a. So this is just a traditional logarithm property. We prove it elsewhere. And we already know it's going to be equal to the right-hand side. It's going to be equal to log base b of x. And now, let's just solve for y. And this is exciting, because y was this thing right over here. But now if we solve for y, we're going to be solving for y in terms of logarithm base b. To solve for y, we just have to divide both sides of this equation by log base b of a. So we divide by log base b of a on the left-hand side, and we divide by log base b of a on the right-hand side. And so on the left-hand side, these two characters are going to cancel out. And we are left with-- and we deserve a drum roll now-- that y is equal to log base b of x divided by log base b of a. So let me write it. Just copy and paste this so I don't have to keep switching colors. So let me paste this. So there you have it. You have your change of base formula. Remember, y is the same thing as this thing right over here. y is log of a. Actually, let me make it clear. y, which is equal to log of a, which is equal to log base a of x-- so copy and paste-- y, which is equal to this thing, which is how we defined it right over here, y is equal to log base a of x, we've just shown, is also equal to this, if we write it in terms of base b. And we have our change of base formula.