Question

Use rules of logarithms to expand log (x+3)^2 / (x-2)(x^2+5)^4

Answer 1

@terenzreignz

Answer 2

That entire thing is under a log, yes?

Answer 3

Yes, the log symbol is before the division

Answer 4

Okay, well, here's the gist of it... logs turn products into sums and quotients into differences...

Here's how it works:
\[\Large \log\frac{ab}{cd}=\log(a)+\log(b)-\log(c)-\log(d)\]
got it so far?
every factor in the numerator gets a plus-sign, every factor in the denominator gets a minus sign.

Use that to your advantage, and tell me what you get...

Answer 5

hmmm but how do I get ab/cd with the exponents

Answer 6

Don't worry about those for now, treat those, say, (x+3)^2 as a single entity/factor for now, we'll deal with those exponents AFTERWARDS :>

Answer 7

So log(x+3) + log(x+3) - log(x-2) - log(x^2+5)?

Answer 8

You forgot the ^4 on the (x^2 + 5)
And I just told you to let those exponents alone :P

Answer 9

But I think you got the idea, sort of... you were close...
\[\Large \log(x+3)^2 - \log(x-2)-\log(x^2+5)^4\]
right?

Answer 10

This is not yet finished, btw... but, do you catch me so far? ^_^

Answer 11

Yes

Answer 12

Okay, NOW we deal with the exponents... using this simple property of logs:
\[\Large \log (t)^\color{red}p= \color{red}p\log(t)\]
Got it?
Recap:
It turns products into sums:
It turns quotients into differences:
It turns exponents into factors ^_^

As you can see, the power can be 'brought down' to be multiplied to the log instead.

Answer 13

2log(x+3) - log(x-2) - 4log(x^2+5)?

Answer 14

Perfect :)
That's all we can do.

Good job ^_^

Answer 15

So that's my answer expanded!?

Answer 16

Yup.

Answer 17

It has the 2 in front? Sometimes I know the 2 would go as a little number on the right of the log

Answer 18

Only the log(x+3) would have a 2, since it had an exponent of 2 earlier.

Answer 19

What you did earlier, which was log(x+3) + log(x+3)
actually accomplishes the same thing, but please don't make a habit of that, and use th exponent-to-factor shortcut ^_^

Answer 20

Ok as long as your sure I got the answer, then thanks! Could you help on one more?

Answer 21

We'll see.

Answer 22

Rewrite using radicals: (a^3/4)^5

Answer 23

an entirely different problem, i see?
Well then, use the laws of exponents first...
\[\Large (a^m)^n = a^{mn}\]

Answer 24

a^3/4 * 5

Answer 25

Which is...?

Answer 26

Multiply fractions... elementary... no?
\[\Large 5\times \frac34 = ?\]

Answer 27

3.75

Answer 28

Please keep it as a fraction...

Answer 29

15/4

Answer 30

Okay, good...
\[\LARGE = a^{\frac{15}4}\]
now, use this fact:
\[\LARGE a^{\frac{\color{red} m}{\color{green} n}}= \sqrt[\color{green} n]{a^\color{red}m} \]

Answer 31

4sqrt a^15

Answer 32

Okay great... that should do it... though it CAN be simplified further.

Answer 33

So that isnt my answer?

Answer 34

It does get the job done, though... but ideally, the exponent inside the radical shouldn't be bigger than the index of the radical [4].

So rewrite it as
\[\Large \sqrt[4]{a^{15}}=\sqrt[4]{(a^3)^4a^3}\]
And I'm sure you can see where this is going...

Answer 35

So should I leave the 4sqrt a^15 as my answer though?

Answer 36

I don't know... it doesn't really say simplify, does it...

Answer 37

No it doesn't

Answer 38

Well then, go for it, and see how it plays out ^_^

Answer 39

haha ok thanks! That is all for now