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# Logarithmic Product

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There are a few rules that can be used when solving logarithmic equations. One of these rules is the logarithmic product rule, which can be used to separate complex logs into multiple terms. Other rules that can be useful are the quotient rule and the power rule of logarithms. The logarithmic product rule is important and is used often in calculus when manipulating logs and simplifying terms for derivation.

The product rule of logarithms. So what I want to do is look at what happens when we are adding up a couple of logs. Okay? So log base 2 of 4 is basically saying what power of 2 will get me 4, so that's 2. Log base 2 of 8 is saying what power of 2 will get me 8 which is 3. So 2+3 this is going to be equal to 5.

What I want to take a look at is what happens when we combine these 2 together. And to get that reference, I'm going to multiply these two insides and put that inside of a log. So log base 2 of 4 times 8, is the log base 2 of 32. Now this is saying what power of 2 will give me 32. 2, 4, 8, 16, 32 fifth power. So what the product rule of logarithms is, is basically saying if we have 2 things inside of a log namely log base b of x times y times and by log base b, this holds for any base as long as the base is a positive number. This is going to be equal to log base b of x plus log base b of y, okay? So this is the product rule of logarithms.

Now if we are multiplying inside of the log, we can split it up as addition outside of the block. Careful thing I want to point out is that this is not the same thing for log base b of x plus y, okay? If we're adding inside the log there is nothing we can do with that. This is stuck as this, okay? It's only working when we're multiplying inside the log.

So, this works both ways if we have the sum of 2 logs. Same base we could put it back together to be a product or if it's a product inside the log we can split it up into 2 different logs.

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