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## Algebra 2

### Course: Algebra 2>Unit 8

Lesson 4: The change of base formula for logarithms

# 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?

• 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? • 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.
• How can logs be applied in something like word problems? Does anyone have an example? • 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.
• So you can introduce a logarithms into an equation and treating it like an algebraic term? • Can I prove this just by trying some numbers? • 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.
• Why can you take logs both sides ? Similarly write both sides as an exponent ? Is both sides equal ❗ • 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.
• 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. • 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? • At of the video, Sal introduces the log_b to both sides. Why is this logarithm needed all of a sudden?   