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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.

• It's almost like you say, because it's so simple that the proof seems unnecessary:

Since you add exponents of powers of the same base to get the exponent of the product, and since logs are exponents, the log_b of the product is the sum of the logs_b of the factors.
• Though it has not been mentioned in the video , what are anti-logarithms?
• An anti-logarithm is essentially exponentiation. For example:

log (base 10) 5 = .6989700043
10^.6989700043 = 5
• Why is the video so blurry?
Pls make another one
• Because it was made it 2008.
• At around , you say that x^l * x^m = x^(l+m)
I thought it would be 2x^(l+m)

WHat am I missing?
• Where would you get the 2 from? You are multiplying, not adding.
Try this with some actual number, maybe that might make the concept more clear:
4² * 4³ = 4*4 * 4*4*4 = 4⁵ = 1024
• Did Sal really refer to his colors 6 times in a single video?
• 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?
• In science and engineering, there are a lot of phenomena that are exponential in nature. The variables are in the exponent, not in the base. In order to solve for the variables you have to take the log of the exponential functions. This is actually used a lot.
• 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