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# Proof of the logarithm product rule

Sal proves the logarithm addition property, log(a) + log(b) = log(ab). Created by Sal Khan.

## Want to join the conversation?

• Either I ams tupid, or Sal is just using the rule as its own proof.

It's like saying 2+2=4, and since 2+2=4, we can deduct 4=2+2, thus 2+2=4. • Though it has not been mentioned in the video , what are anti-logarithms? • At around , you say that x^l * x^m = x^(l+m)
I thought it would be 2x^(l+m)

WHat am I missing? • Did Sal really refer to his colors 6 times in a single video? • Why is the video so blurry?
Pls make another one • Feels like he did this one on an etch a sketch. Can't read any of that ;-x • how is this concept useful guys somebody help? • I know this is wrong but I can't think out of it.

So if (LogX(A)=l) + (LogX(B)=m) = (LogX(A*B)=n) by the product rule. And if were to be converted to exponential form, would it look like this: (X^l=A) + (X^m=B) = (X^n=A*B), and if it is, then wouldn't this be equivalent to A+B=A*B which doesn't sound right. • Let's go through the correct application of the logarithmic properties and show why the statement is incorrect:

The product rule for logarithms states that log_x(A) + log_x(B) = log_x(A * B).

Suppose we have the expressions: (LogX(A) = l) and (LogX(B) = m).

According to the product rule, combining these two expressions should give us:
log_x(A) + log_x(B) = log_x(A * B).

However, we cannot directly add the two logarithmic expressions (log_x(A) and log_x(B)) as if they were numerical values (like "l" and "m").

To convert to exponential form, we would use the following:

log_x(A) = l -> x^l = A
log_x(B) = m -> x^m = B
Then, we can apply the product rule to the exponential forms:

x^l * x^m = A * B

Using the property x^a * x^b = x^(a+b):

x^(l+m) = A * B

However, we cannot say that (x^l = A) + (x^m = B) = (x^(l+m) = A * B).

Your observation that this would lead to A + B = A * B is indeed correct, but that's because the manipulation of logarithmic expressions in this way is not valid.

It's essential to use logarithmic properties correctly and to remember that logarithms do not follow the same arithmetic rules as regular numbers. When dealing with logarithms and their properties, it's crucial to apply them correctly to avoid incorrect conclusions or statements.

"Never back down, never give up"- Nick Eh30  