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# Proof of the logarithm quotient and power rules

Sal proves the logarithm quotient rule, log(a) - log(b) = log(a/b), and the power rule, k⋅log(a) = log(aᵏ). Created by Sal Khan.

## Want to join the conversation?

• what grade is logs for?
• Logarithms are part of the High School Algebra Core Curriculum Standards
• I think I just gained several brain cells at
• who can prove this to me?? how it can be answer of 1?
1/(log_a AB) + 1/(log_b AB) = 1
• That's easy (but changing b to x since there is a subscript x character):
1/logₐ(ax) + 1/logₓ(ax)
= [ log(a) / log(ax)] + [ log(x) / log(ax) ]
= [ log(a) + log (x) ] / log(ax)
= log (ax) / log (ax)
= 1
Provided that both a and x are positive. It is undefined if either a or x is ≤ 0
• What's the point of proofs?
• The point is to prove that this rules are not made up and that they are true. Just as you could put in the numbers into a formula you could prove it by using the properties o a function/operator to find out how the numbers move.
• Where is the next video Sal mentions at ?
• By knowing the previous two properties (product and power), you could prove the quotient property this way:
log(a) - log(b) = log(a) + (-1)log(b) = log(a) + log(b^-1) = log(a) + log(1/b) = log(a * 1/b) = log(a/b)
• There are more logarithm properties than this, they should be added to this section.
• Indeed there are way more rules, at least the ones I studied at school
• how would you solve: 12^log12(4)
• since 12 is being brought to a log power with a base the same as it, they cancel out. so 12^log12(4)=4 Keep in mind this only works because there are the same number in those two specific places.

To see that it works first set it up like an equation.

12^log12(4) = x

Now turn it into another log. so if it were 12^y = x you sould make it log12(x)=y. so here 12^log12(4)=x becomes log12(x) = log12(4).

x has to be 4 because if log12(x) = log12(4) log base 12 only gets to the number log12(4) equals when the number inside is 4, no other number can get the sam result. so x must be 4
• Can anyone prove that logaX=-logaX using log law 3?