Logarithms

Purple Gorilla

Anyone good at logarithms? We just started and I'm just not getting them. We have to do a few for homework and I only got one

Like what do you do when you have a log on the other side aswell like in question 15? Also, what do you do when it's a log minus a log like in 17?

Thanks

LeixlipRed

You need to revise your rules of logarithms as seen here

Purple Gorilla

Thanks. I know the rules of the Logarithms but I just can't seem to apply them. I've spent about 40 minutes looking at these and I just can't get them to work Like on question 17 I do it so it's
Log2(X2-1)=Log23

(The bolded 2 is meant to be an index)
but I don't know where to go from there

LeixlipRed

You mean question 15? You can equate [latex]x^2 - 1[/latex] and 3 and solve for x because both are logs to the same base.

Purple Gorilla

Thank you for that!!!

Fremen

Put the work in now, Now, NOW.
Unless you do a philosophy degree, these will come up in college. Five hours of work now will save you fifty hours of misery later. Forget the quadratic formula, Sin, Cos, and Tan. These are what you will use later on in life.

In each case, you have an expression of the form Log(x) (operation) Log(Y) = constant. Use the rules for logs to express that in the form Log(z) = constant, then raise both sides to the base of the log.

I was a bit confused by them when I first saw them. I found it helped to write the problem out in "essay form". Then [latex]\log_2 (x) = c[/latex] becomes
"What power do I have to raise 2 to in order to get c?".

[latex]2^{\log_2 (x)}[/latex] becomes "Two raised to the power that I need to raise two to in order to get X". (That's X, of course).

Purple Gorilla

Ok I'm back...what do I do if there's a a fraction in the equation and there are different bases?
The question is log5X= 1/logx5

I brought it all over to one side so it's all equal to zero but how to I eliminate the fraction?

LeixlipRed

That link I gave you previously has a formula for changing bases. Have a look.

Purple Gorilla

Thanks for that. I did it and I keep getting X=1 as the answer but the back of the book says X=25?
What I did was:
Bring it all to one side and = 0, change base

Then to change the base I did Log55/Log5X
. To get rid of the fraction I multiplied the top by the bottom and I got Log55X

I was left with Log55x-1=0 so I brought the 1 to the other side and then did 5^1=5X therefore X=1.


Did I go completely wrong?

LeixlipRed

That one might not be as straightforward as I first thought. 25 is the answer but to find it the change of base formula won't work so simply. Dunno how you got [latex]log_{5}5x[/latex] though! That's wrong.

Purple Gorilla

I'm not sure how to multiply Logs Ugh I just don't get these

Purple Gorilla

Would you be able to show me how to do it but show it step by step so I know what to do? I just have no clue how to change the base because the example he gave us where you change the log had a number before the log so it would be like 2log5 if you get me? Then when he did that fraction to change the base, he just substituted the bottom of the fraction with a 2 and then cancelled out the two 2s..you can't do that in this one because there's no number before the log and it's a fraction

LeixlipRed

Hmmm, I think I see where my confusion is. Is the question:

[latex]log_{5}x = {1}/{log_{x}5}[/latex]

or

[latex]log5x = {1}/{log_{x}5}[/latex]

The first one is true for any x and that's what I was working on.

Purple Gorilla

The 1st one I believe. The question says:

Show that log5X= 1/logx5 and hence solve the equation 9logx5= log5x

I'm not worrying about the 2nd part yet. If I'm able to understand the first part then I'd like to try the 2nd part on my own to see if I really get it!

I've tried literally so many different combinations of how to do the question and I keep coming up with X=1 ..it always comes down to 5X=5, X=1

Purple Gorilla

Woops..didn't see the picture thingy properly.. It's the 1st one

LeixlipRed

Hmmm, are you sure? Because the question ask you to "show" that the first thing is true. Not solve for x. I think you need to start with the left hand side which is [latex]log_{5}x[/latex]. Use the change of base formula to prove this equals [latex]1/{log_{x}5}[/latex]. Then use that fact to solve the second expression for x.

LeixlipRed

Ok, well that's definitely what you need to do. You cannot solve for x in the first expression as it's true for all x.

Purple Gorilla

So would I change the base on the left hand side to match the base of the fraction?

Just tried it and I'm not sure if I'm doing it right. To change the base I did logxX/logx5

Is that right? Doesn't look like it is

LeixlipRed

It is right. What is [latex]log_{x}x[/latex]? It's the number you must raise x to to get x. Or [latex]x^?=x[/latex]. What is that number?

Purple Gorilla

I don't know.

X^1=x ?

LeixlipRed

Yes. So what have you just shown then?

Purple Gorilla

I'm still a tiny bit confused but does it mean logxX^1/logx5= 1/logx5 ?? So I've just proven that it all equals each other?

LeixlipRed

You've shown that:

[latex]log_{5}x =\frac {log_x{x}}{log_x{5}}[/latex]

And because, as we discussed, [latex]{log_x{x}} = 1[/latex] you've shown that:

[latex]log_{5}x =\frac {1}{log_x{5}}[/latex]

Sheeps

Fremen wrote: »

Forget the quadratic formula, Sin, Cos, and Tan.

Eh... ?

Purple Gorilla

Thanks so much for that leixlip. I actually get it now!!!! Thank yooooouuuuuuuu

LeixlipRed

Remember to do the second part though

Fremen

Sheeps wrote: »

Eh... ?

I meant in terms of relative importance, not in absolute terms.

Sheeps

Fremen wrote: »

I meant in terms of relative importance, not in absolute terms.

Well it depends entirely on what field you're going in to. For example, computer graphics. Knowing about Logarithms is only really important for understanding algorithmic complexity. The core mathematical skills you will be using will be trig, vector math and matrices.

Fremen

For sure. But the average man on the street won't use that stuff.
Clearly an engineer (for example) will use a whole load of different tools. My point was that if you "average" over all professional fields, logs and exponents are probably the mathematical tools which will crop up most often.

Edit: Having said that, I've done a good six years of undergraduate and postgraduate maths. Haven't used the change of base formula once.