Main content

## Algebra 2

### Course: Algebra 2 > Unit 8

Lesson 4: The change of base formula for logarithms- Evaluating logarithms: change of base rule
- Logarithm change of base rule intro
- Evaluate logarithms: change of base rule
- Using the logarithm change of base rule
- Use the logarithm change of base rule
- Proof of the logarithm change of base rule
- Logarithm properties review

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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.

## Want to join the conversation?

- How can logs be applied in something like word problems? Does anyone have an example?(7 votes)
- 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.(18 votes)

- At2:38, 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?(9 votes)- 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.(13 votes)

- 2:32So you can introduce a logarithms into an equation and treating it like an algebraic term?(6 votes)
- You bet! As long as you do the same thing to both sides you can do logs, or exponentials, differentiation and integration, anything you think will make it easier for you to solve the problem.(12 votes)

- Can I prove this just by trying some numbers?(2 votes)
- 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.(11 votes)

- 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?(2 votes)
- Yes. If you have a log button with no options to change the base number, it is usually base 10.

This is because most of what we do in our everyday life is in the base 10 system, like counting.(3 votes)

- Why can you take logs both sides ? Similarly write both sides as an exponent ? Is both sides equal ❗(2 votes)
- 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.(3 votes)

- 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.(1 vote)- I mean, you're in one of the world's biggest learning platforms, so maybe try typing this into the "Search" in Khan Academy.(4 votes)

- At3:04, why is logarithm base b of a to the y is the same as y times the logarithm base b of a?(2 votes)
- This is a real and very correct property for logarithms, which Sal calls the power rule. He explains and proves it in the unit about logarithms, in the section called properties of logarithams. Hopefully you can understand after you see the videos about this property.(2 votes)

- 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.(2 votes)
- I understand based on a previous question that log base 1 is undefined.

However, another way log base b of A could equal zero is if A = 1.

Shouldn't we make sure that's not the case before dividing by log base b of A?(1 vote)- In the change of base formula, the log of A would be in the numerator, not the denominator. The log base b of 1, such that b is greater than one, according to the change of base formula is equal to log base c of 1 divided by log base c of b. Log base c of 1 equals zero, and log base c of b is nonzero because b does not equal one, so log base c of 1 divided by log base c of b is equal to zero, as is expected.(2 votes)

## 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.