Math Is Fun Forum

Whizzies

Member

Registered: 2014-07-18

Posts: 53

Logarithm which goes first?

Hello fellow people,

question:

I have learned this:

log (uw) = log u + log w
log (U/W) = log u - log w
log U^2 = 2 log u

My question is this:

2 * log2 (x+3) = log2 (12) - log2 (3) =

log2 (x+3)^2 = log2 (4)
(x+3)^2 = 4

What must go first? Multiply addition or subtraction?

Like 2 * 3 + 4= x, the rules are 2*3 first than add 4.. but how does it work with logaritms?

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: Logarithm which goes first?

You’ve got as far as

[list=*]
[*]

[/*]
[/list]

Now just solve it for x.

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Bassaricyon neblina

Whizzies

Member

Registered: 2014-07-18

Posts: 53

Re: Logarithm which goes first?

You’ve got as far as

[list=*]
[*]

[/*]
[/list]

Now just solve it for x.

Yes, very true. But in the future I have to solve lots and lots of equations, and I'd like to know the basic rules (:

Another question though!

x^2+6x+5 = 0
(x+5)(x+1)=0
x = -5 and x = -1

it does not work for x = -5. Why do we check this with the first formula? (log x+3) and not with the later formula ((x+3)^2)?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Logarithm which goes first?

Why does it not work for x = - 5?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Whizzies

Member

Registered: 2014-07-18

Posts: 53

Re: Logarithm which goes first?

Why does it not work for x = - 5?

Log is only defined for x>0

log (x+3)
x = -5
log (-5+3) = log (-2)
Log (-2) is not defined.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Logarithm which goes first?

hi Whizzies,

You are wise to check your answer back in the original question.

-5 is a solution to the quadratic.  Does that mean it must be a solution to the log equation ?

First consider this:

A rectangle has a length that is 2 bigger than its width.  The area is 24.  Find the length and width.

Let the width be x  => length = x + 2 so

area = 24 = x(x+2)     =>     x^2 + 2x - 24 = 0 = (x-4)(x+6)

So x = 4 or -6.

Now width = 4 and length = 6, is a reasonable solution.  But what about width = -6 and length = -4 ?

As -6 x - 4 = 24, you might accept this uncritically as another answer.  But surely we should object to having negative distances.

If you wanted to be bold, you could invent a whole new branch of mathematics where negative distances are allowed.  But you'd have trouble drawing this rectangle.  It all depends on how the original question was set.  Was the questioner expecting only real number solutions ?  If so, then reject x = -5.  But read on:

The 'laws' for logs of negatives work the same as the laws for logs of positives.

eg.

And the definition of logs has been extended to cover this sort of thing:

http://en.wikipedia.org/wiki/Complex_logarithm

So, keep up the good work and keep checking those answers. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Logarithm which goes first?

Yep, Bob is right, check, check, check.

Log is only defined for x>0

Although I do not believe that is true I thought you meant that x = - 5 was not working as a solution to the quadratic. Sorry for that confusion.

Ain't a quadratic the long way. Why not?

Clean up first.

No quadratics and no spurious roots.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: Logarithm which goes first?

There seems to be some confusion here. What Whizzies means is that (excluding complex numbers) log(y) is only defined for y > 0. The problem involves log(x+3), so y = x+3. It is fine for x = −1; it is the y that needs to be > 0.

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Bassaricyon neblina