If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Intro to logarithm properties (2 of 2)

Sal introduces the logarithm identities for multiplication of logarithm by a constant, and the change of base rule. Created by Sal Khan.

## Want to join the conversation?

• At the very end of the video, how did you get 5/2 from "1/2 log base 2 of 32"? And how did you get 3/4 from "1/4 log base 2 of 8"?
• The expression "log base 2 of 32" is equal to 5 (because 2 to the 5th power equals 32).. Sal didn't write that step but then he just multiplied 5 * 1/2 to get 5/2. He the did the same thing to get 3/4 from "1/4 log base 2 of 8"... (2 raised to the 3rd power = 8, so log base 2 of 8 = 3)... then he just multiplied 3 * 1/4 to get 3/4. Hope that helps a bit! Logs took me a while to understand, but after some practice they're not so bad.
• I don't quite understand this part though, but anyway where did you get the 1/2 from at ???
• Hey, Rizky. He got the 1/2 because that's one of the rules. If you have a square root, you can get rid of the radical sign and make it a power of 1/2. So, that's all he did was raise the quantity under the sign to 1/2 power and got rid of the square root sign. This works for all roots. If it was a cubed root, we could get rid of the sign and raise the quantity to 1/3 power. If it was a fourth root, we could get rid of the sign and raise the quantity to 1/4 power and so on.
• The exercise "Logarithms 2" is asking me to find log(3) + log(5).
This confuses me because there's no number above them, and because they have different bases. Can someone please explain to me why this is?
• Hello Meredith. When "log" is written without subscripts (little numbers below the word log) it is assumed to be base 10. The 10 is left out. (Like the positive sign in positive numbers). So here you are adding two logarithms with base 10. As Sal explains log(a)+log(b) = log (a times b). The answer here is log(15).
• I must have missed a lesson somewhere along the line, why is it that the fraction, 32 over the square root of 8, is formed into a subtraction question not a division question?
• 2*2*2*2*2*2*2*2*2*2 divided by
2*2*2*2*2
Five of the twos cancel each other, leaving five twos remaining in the numerator. Hence, 2^10 / 2^5 = 2^(10-5) = 2^5
Hope this helps explain that exponent rule.
• I hate memorizing. I love understanding. Please someone help me understand how did the exponent convert into a coefficient, and vice-versa. Thanks
• log_b (x^e) = y [Let]
So, x^e = b^y
So, (x^e)^(1/e) = (b^y)^(1/e) = b^(y/e)
So, x = b^(y/e)
So, log_b (x) = y/e
So, e log_b (x) = y = log_b (x^e)

This is the formal proof.

Taking a simpler example using a previously learnt property,

log_b (x^5)
= log_b (x.x.x.x.x) = log_b (x) + log_b (x) + log_b (x) + log_b (x) + log_b (x)
= 5 log_b (x)

and 5 is the original exponent of x. Hence proved.
• I lost him where he appears to have reduced 1/2log2,32 to 5/2, at . Help.
• This is because the 1/2 cancels out the outer square root, but it doesn't affect the square root of 8 because the 8 is a square root inside the outer square root.
• At , what does C stand for?

Those are awesome videos. Keep making more.
• It is simply a variable for the possible number that can go there.
• also, do we always assume that "C" in the 2nd property of this video will be base 10 or can we just throw any number in for "C"?
• It can be any number however if you have a calculator it will have a log base 10 button built in which makes using 10 as C easier.
• how to solve: log base 3 of 9x^4 - log base 3 of (3x)^2
• Firstly, you cannot solve an expression, but you can simplify it like this:
log base 3 of 9x^4 - log base 3 of 9x^2 =
log base 3 of (9x^4/9x^2) =
log base 3 of (x^4/x^2) =
log base 3 of (x^2)