Given the parent functions f(x) = log2 (3x − 9) and g(x) = log2 (x − 3), what is f(x) − g(x)?

Question

anonymous · Answer

@e.mccormick

mathstudent55 · Answer

Subtract the functions.

mathstudent55 · Answer

$\Large f(x) = \log_2 (3x - 9)$

$\Large g(x) = \log_2 (x - 3)$

$\Large f(x) - g(x)  = \log_2 (3x - 9) - \log_2 (x - 3)$

Ok so far?

anonymous · Answer

yes @mathstudent55

mathstudent55 · Answer

Now we need to simplify the fright side using some rules of logs.

Here are two rules of logs:

$\Large \log ab = \log a + \log b$

$\Large \log \dfrac{a}{b} = \log a - \log b$

anonymous · Answer

Alright.

anonymous · Answer

Ok, i thought we were going to factor lol

mathstudent55 · Answer

Yes, we do factor. Look at the first log below compared to what it was.

$\Large f(x) - g(x)  = \log_2 3(x - 3) - \log_2 (x - 3)$

anonymous · Answer

you moved the 3 out of (3x-9) so u divide by 3

mathstudent55 · Answer

I only factored out the 3. 3(x - 3) = 3x - 9. We are still taking the log of the same quantity, only it is now written in a different form, its factored form.

Now we use the first rule of logs above to deal with the first log in our equation.

mathstudent55 · Answer

I'm now working on the red part:

$\Large f(x) - g(x)  = \color{red}{\log_2 3(x - 3)} - \log_2 (x - 3)$

$\Large f(x) - g(x)  = \color{red}{\log_2 3 + \log_2 (x - 3)}  - \log_2 (x - 3)$

You see how the rule of logs is applied?

anonymous · Answer

Yes :)

mathstudent55 · Answer

What can you do to the green part below?

$\Large f(x) - g(x)  = \log_2 3 + \color{green}{\log(x - 3) - \log_2 (x - 3)}$

anonymous · Answer

Cancel out?

anonymous · Answer

would the answer be log 2 1/3

mathstudent55 · Answer

Correct.

mathstudent55 · Answer

$\Large f(x) - g(x)  = \log_2 3 $

anonymous · Answer

is it log 2 3 or log 2 1/3?

mathstudent55 · Answer

Answer is: 

$\Large f(x) - g(x)  = \log_2 3 $

anonymous · Answer

Alright, Thanksss! :)

mathstudent55 · Answer

You're welcome.

anonymous · Answer

Given the parent functions f(x) = log10 x and g(x) = 5x − 2, what is f(x) • g(x)?
for this one, i got  f(x) • g(x) = log10 x5x − 2

anonymous · Answer

Was I right?

mathstudent55 · Answer

Here you need to multiply a log by a binomial.

$\Large f(x) = \log_{10} x$

$\Large g(x) = 5x − 2$

Multiply the left sides together and multiply the right sides together:

$\Large f(x) \cdot g(x) = \log_{10} x \cdot (5x - 2)$

Use the commutative property on the right side:

$\Large \color{red}{f(x) \cdot g(x) = (5x - 2)\log_{10} x} $

Now use the distributive property:

$\Large \color{green}{f(x) \cdot g(x) = 5x\log_{10} x - 2 \log_{10} x}$

mathstudent55 · Answer

You were close. The idea was correct, but you need to use parentheses.

mathstudent55 · Answer

Any of the lines above is a correct answer, but to make the line in black clearer, I think it's better to write it as:

$\Large f(x) \cdot g(x) = (\log_{10} x) (5x - 2)$

anonymous · Answer

Okay! Thanks :)