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# 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.

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• 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 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)
• Oof he kinda lost me with that last example

Great videos though! Really appreciate it
• Assuming you are referring to the example
log_2(sqrt(32 / sqrt(8)))

There are different ways to do this, but I will follow the way the video described.

First, you need to know the following 2 formula / property
sqrt(a) = a^(1 / 2)
Simply definition of square root.

log(a^n) = n * log(a)
Mentioned at the beginning of the video.

Now we can rewrite our initial equation as
log_2((32 / sqrt(8))^(1 / 2))
= (1 / 2) * log_2(32 / sqrt(8))

Then, they make use of the following log property
log(a / b) = log(a) - log(b)
Mentioned in the last video Part 1.

We can continue our calculation.
(1 / 2) * log_2(32 / sqrt(8))
= (1 / 2) * (log_2(32) - log_2(sqrt(8)))
= (1 / 2) * (log_2(32) - log_2(8^(1 / 2)))
= (1 / 2) * (log_2(32) - (1 / 2) * log_2(8))
= (1 / 2) * (5 - (1 / 2) * 3)
= 1.75

If you still have any question, feel free to ask.