Question

Can someone please help with this trig question?

Answer 1

First we must understand how to write a logarithmic function:

$log_{b}a=x$

In the equation above, b is the base, x is the exponent, and a is the answer. These same variables can be rearranged to be expressed as an exponential equation as followed:

$b^x=a$

Next, we need to understand basic logarithm rules.

1. When a value is raised to a power, we can move the exponent to the front of the logarithm. Example:

log(a^2) = 2log(a)

2. When two variables are multiplied together, we can add the logarithms of the individual variables together. Example:

log(ab) = log(a) + log(b)

3. When a variable is divided by another variable, we can subtract the logarithms of the individual variables. Example:

log(a/b) = log(a) - log(b)

Now we can use these rules to solve the problem.

$log(r)=log( \sqrt[3]{ \frac{A^2B}{C} } )$

We can rewrite the cube root as:

$log(r) = log( (\frac{A^2B}{C})^ \frac{1}{3} )$

Now we can move  the one-third to the front:

$log(r) = \frac{1}{3} log( \frac{A^2B}{C} )$

Now we can split up the logarithm:

$log(r) = \frac{1}{3} (log(A^2)+log(B)-log(C))$

Finally, we can move the exponent to the front of the log of A:

$log(r) = \frac{1}{3} (2log(A)+log(B)-log(C))$

Distribute the one-third to get the answer:

$log(r) = \frac{2}{3} log(A) + \frac{1}{3} log(B) - \frac{1}{3} log(C)$

The answer is (4).