Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2015-05-10 03:35:10

Nemexia
Member
Registered: 2015-05-10
Posts: 13

log(a,x)=a^x

we have two functions:

f(x)=log(a,x)
g(x)=a^x

a=??

in what "a" the graph of these two hit each other in only one point?


i think it has the same meaning as:

in what "a" the following equation has only one answer:

log(a,x)=a^x

Offline

#2 2015-05-18 00:28:35

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: log(a,x)=a^x

hi Nemexia

What do you mean by f(x)=log(a,x)

Is this

?

Bob


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 smile

Offline

#3 2015-05-19 00:43:27

Nemexia
Member
Registered: 2015-05-10
Posts: 13

Re: log(a,x)=a^x

a is base

Offline

#4 2015-05-19 01:44:38

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: log(a,x)=a^x

hi Nemexia

Thanks for clearing that up.  I'm getting a value of about a = 1.445 but only by looking at the graphs.  I will try to find a better method.  Here is what I did:


and

Then I used http://www.mathsisfun.com/data/function-grapher-old.php

to get these:

kBNCKRl.gif

Here you can see the graphs when a = 1.2 and when a = 1.5

I tried different a values until the graphs just touched.  I'll keep thinking and try to come up with a way to calculate this a exactly.

LATER EDIT:

These functions are inverses of each other so we want the graph where the line y=x makes a tangent to the curve.  ie gradient angle = pi/4.

Got to go out now, so I'll finish this off when I get back.

Bob

Last edited by Bob (2015-05-19 01:58:25)


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 smile

Offline

#5 2015-05-19 02:51:38

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

Re: log(a,x)=a^x

Hello:

Acronyms are for people who can't spell.

Let's stop writing EM. sad

Last edited by ElainaVW (2015-05-19 03:05:38)

Offline

#6 2015-05-19 03:02:22

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: log(a,x)=a^x

hi ElainaVW

Thanks.  I just logged in to say that as well.  Except I don't know why.  I've just got it by trial.

Bob


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 smile

Offline

#7 2015-05-19 03:20:09

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

Re: log(a,x)=a^x

You had to make an ansatz too! smile

Offline

#8 2015-05-19 19:01:36

Nemexia
Member
Registered: 2015-05-10
Posts: 13

Re: log(a,x)=a^x

yeah! , thanks.
Can you say how you found it?

Offline

#9 2015-05-20 19:15:36

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: log(a,x)=a^x

From the graphs (using trial to 'home in' on the value of a) it looked like this single point of contact was at (e , e)  so I used differentiation to get an expression for the gradient and set this equal to 1.  Then using (e , e) I was able to get 'a'.  Substituting back shows this value is right which is what ElainaVW meant by 'ansatz'.  I cannot find an algebraic way to derive a value for a.

Bob


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 smile

Offline

Board footer

Powered by FluxBB