Select the correct answer.
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At first glance, this problem seems cumbersome, but the great thing about the properties of logarithms is that they allow us to break our problem down into chunks and sort through each on its own.
Let's look at what we've been given:
![\log_3{(x^{20}\cdot \sqrt[3]{y^6} )](https://tex.z-dn.net/?f=%5Clog_3%7B%28x%5E%7B20%7D%5Ccdot%20%5Csqrt%5B3%5D%7By%5E6%7D%20%29)
We can start by using the property that
to separate our single logarithm into the sum of two:
![\log_3x^{20}+\log_3\sqrt[3]{y^6}](https://tex.z-dn.net/?f=%5Clog_3x%5E%7B20%7D%2B%5Clog_3%5Csqrt%5B3%5D%7By%5E6%7D)
Next, let's simplify the term![\sqrt[3]{y^6}](https://tex.z-dn.net/?f=%20%5Csqrt%5B3%5D%7By%5E6%7D%20)
into something a little more workable. We can turn the radical ![\sqrt[3]{\ }](https://tex.z-dn.net/?f=%20%5Csqrt%5B3%5D%7B%5C%20%7D%20)
into the rational exponent 1/3, which lets us obtainin a new form for the y term:
We now have:
We can now use the property that
to bring the exponents on the x and the y out front and obtain our final answer:
Where 20 is our A and 2 is our B
Let's look at what we've been given:
We can start by using the property that
Next, let's simplify the term
We now have:
We can now use the property that
Where 20 is our A and 2 is our B