Math Is Fun Forum

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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

Nemexia

Member

Registered: 2015-05-10

Posts: 13

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

a is base

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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:

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

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.

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

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

You had to make an ansatz too!

Nemexia

Member

Registered: 2015-05-10

Posts: 13

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

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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