Question

How do I solve this promblem

Answer 1

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:

$\log_3{(x^{20}\cdot \sqrt[3]{y^6} )$

We can start by using the property that $\log_n(a\cdot b) = \log_na+\log_nb$ to separate our single logarithm into the sum of two:

$\log_3x^{20}+\log_3\sqrt[3]{y^6}$

Next, let's simplify the term $\sqrt[3]{y^6}$ into something a little more workable. We can turn the radical $\sqrt[3]{\ }$ into the rational exponent 1/3, which lets us obtainin a new form for the y term:

$y^{6(1/3)}=y^{6/3}=y^2$

We now have:

$\log_3x^{20}+\log_3y^2$

We can now use the property that $\log_na^x=x\log_na$ to bring the exponents on the x and the y out front and obtain our final answer:

$\log_3x^{20}+\log_3y^2=20\log_3x+2\log_3y$

Where 20 is our A and 2 is our B