Inverse of Logarithm: Understanding ax

The inverse function takes us back where we started, sof^{-1}( f(x) ) = x.\begin{align*}f(x) & = \log_a x\\f^{-1}(x) & = a^x\\f^{-1}( f(x) ) & = a^{\log_a x} = x\end{align*}In summary, the conversation discusses the inverse function of a logarithm, which can be represented by the equation f(x) = logax. The inverse function maps
  • #1
nobahar
497
2
Hello!
Run into trouble...again.
This concerns the inverse function of a logarithm
If a function maps x on to logax, then the inverse maps logax on to x.
So, f(x) = logax, can be presented as y=logax; therefore, x=ay.
The book states that the inverse is ax, why is this the inverse?
I tried determining the inverse by using a basic inverse:
f(x)=3x+1
f-1(x)=(x-1)/3
If x=2
Then, f(x)=3*2+1=7
and f-1(x)=(7-1)/3=2
If the product of logax=y, going the other way involves ay=x, yet the inverse is, apparently, ax.
I don't see why this is so...

P.S. Happy voting!
 
Physics news on Phys.org
  • #2
So, f(x) = logax, can be presented as y=logax; therefore, x=ay.

You are correct. I think the book is just redefining "x". Kind of the way you did when you used x in both the function and inverse function definitions...

f(x)=3x+1
f-1(x)=(x-1)/3

It's better to be explicit about a different "x" and "y" for the two different directions...
 
  • #3
Your work is correct; the difference is mostly a matter of notation.

One old1 trick for finding inverses of functions is to begin by reversing the roles of [tex] x [/tex] and [tex] y [/tex] at the start. The complete process would look like this.

[tex]
\begin{align*}
f(x) & = \log_a x \\
y & = \log_a x \tag{Now switch variables}\\
x & =\log_a y\\
a^x & = y,
\end{align*}
[/tex]

so the inverse function has equation

[tex]
y = a^x
[/tex]

We are accustomed to writing functions in the [tex] y = [/tex] form in early math courses.

Footnote 1: I know this is an old procedure because I learned it when I was in high school: if it has been around that long, it is indeed old.
 
  • #4
Thanks for the quick replies! The responses are much appreciated, and even tickled me :smile:, not something one would associate with a maths help forum!
So ax is really ay 'in disguise', since it is the inverse function the y is now the x?
 
  • #5
Don't write "x" and "y" in isolation. That is what is confusing you. The functions f(x)= ln(x) or f(y)= ln(y) or f(z)= ln(z) are all exactly the same function.

If f(x)= ln(x) then f-1(x)= ex or f-1(y)= ey or f-1(a)= ea, etc. it doesn't matter what you call the variable.
 
  • #6
Okay, I thought I had it...
If f(x)=y, then f(x)=logax, and ay=x, and so af-1(x)=x?
If f(x)=logax and the inverse is ax, and I attempt an equation it doesn't work... x has to change going in the different 'ends', and so x in the new equation (the inverse) is y on the first equation...
I appreciate this may be getting tiresome to explain... but please bear with me! :smile:
Thanks!
 
  • #7
Sure it does
[tex]a^{log_a x}= x[/tex]
and
[tex]log_a(a^x)= x[/tex]

Once again, f(x)= loga(x), f(y)= loga(y), f(z)= loga(z), etc. are all exactly the same function.
 
  • #8
The notation is getting to you.

[tex]
\begin{align*}
f(x) & = \log_a x\\
y & = \log_a x \\
x & = \log_a y\\
a^x & = y
\end{align*}
[/tex]

This means that the function [tex] f(x) = \log_a x [/tex] has as its inverse

[tex]
f^{-1}(x) = a^x
[/tex]
 

FAQ: Inverse of Logarithm: Understanding ax

What is the inverse of a logarithm?

The inverse of a logarithm is the exponent that must be raised to a given base to obtain a specific number. This can also be thought of as "undoing" the logarithm operation.

How is the inverse of a logarithm expressed mathematically?

The inverse of a logarithm is typically written as logb-1(x) or bx, where b is the base and x is the number being raised to the power.

What is the relationship between logarithms and their inverses?

The inverse of a logarithm is closely related to the logarithm itself. They are essentially opposite operations. For example, if logb(x) = y, then logb-1(y) = x.

How do you find the inverse of a logarithm?

To find the inverse of a logarithm, you can use the exponentiation property of logarithms. This states that logb-1(x) = bx. Alternatively, you can use the change of base formula to rewrite the logarithm in a different base and then solve for the exponent.

Why is understanding the inverse of logarithms important?

Understanding the inverse of logarithms is important because it allows you to solve exponential equations and perform other mathematical operations involving logarithms. It also has many real-world applications, such as in finance, physics, and computer science.

Similar threads

Replies
7
Views
1K
Replies
15
Views
1K
Replies
3
Views
1K
Replies
2
Views
1K
Replies
19
Views
3K
Replies
6
Views
2K
Back
Top