Simplify the expression
1 answer:
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Answer:
Step-by-step explanation:
So, there are two logarithmic identities you're going to need to know.
<em>Logarithm of a power</em>:
So to provide a quick proof and intuition as to why this works, let's consider the following logarithm:
Now if we raise both sides to the power of c, we get the following equation:
Using the exponential identity:
We get the equation:
If we convert this back into logarithmic form we get:
Since x was the basic logarithm we started with, we substitute it back in, to get the equation:
Now the second logarithmic property you need to know is
<em>The Logarithm of a Product</em>:
Now for a quick proof, let's just say:
Now rewriting them both in exponential form, we get the equations:
We can multiply a * c, and since b^x = a, and b^y = c, we can substitute that in for a * c, to get the following equation:
Using the exponential identity: , we can rewrite the equation as:
taking the logarithm of both sides, we get:
Since x and y are just the logarithms we started with, we can substitute them back in to get:
Now let's use these identities to rewrite the equation you gave
As you can see, this is a log of products, so we can separate it into two logarithms (with the same base)
Now using the logarithm of a power to rewrite the log(x^2) we get:
