84
votes
Accepted
Why are there two inverses to exponentiation?
The concept of an inverse operation itself is a bit tricky. Often we consider arithmetic operations to be binary operations:
$\DeclareMathOperator{\add}{add}$
$\DeclareMathOperator{\subtract}{subtract}...
- 3,160
33
votes
Why are there two inverses to exponentiation?
Is your variable at the base or the exponent?
An exponential function is a function of the form $f(x)=a^x$ for some constant $a$. In this case, the inverse is indeed given by a logarithm $f^{-1}(x)=\...
- 769
23
votes
Why are there two inverses to exponentiation?
(Despite good answers already, I thought the concrete example below could be useful.)
For multiplication and addition, there is exactly one inverse operation, namely division and subtraction.
Yet ...
- 399
10
votes
Accepted
Apply the inverse operation on both sides, or know the inverse function?
I'd say both: they must know the inverse function, and that it's being applied to both sides. This then connects up with the fundamental properties of equality. As a teacher of many algebra courses (...
- 26.6k
6
votes
Accepted
What are some common ways students get confused about finding an inverse of a function?
I've noticed a few issues when students solve problems of the form, "Find the inverse of this function", and not all of the issues are necessarily because of the students' misunderstanding of what an ...
- 10.9k
4
votes
Why are there two inverses to exponentiation?
The other answers have shown that it depends on which of the two inputs you fix and which you let vary. As a way to visualize this, here's a plot of the 3D surface $z = x^y$. I've drawn two polynomial ...
- 141
4
votes
Accepted
Why many people believe that: $\displaystyle c>0\implies \frac{1}{c}<0$?
I don't think anyone believes this in the way you have stated. Perhaps you should be more concrete in how the student is actually being presenting the misconception. Perhaps they think $\frac{1}{2} &...
- 419
3
votes
What are some common ways students get confused about finding an inverse of a function?
From a comment by the OP:
I'm trying to come up with "plausible" wrong answers for a multiple choice question about finding inverses.
Per an answer given to this question, you might be able to ...
- 9,719
3
votes
Why bother calculating the inverse of 3 by 3 matrix?
In computer graphics, the view matrix is the inverse of the camera matrix.
This is needed, e.g., in game programming.
In general, matrices are used to convert from coordinate system A to coordinate ...
- 30.1k
2
votes
Apply the inverse operation on both sides, or know the inverse function?
The fundamental concept is always: You can do the same thing to both sides, as long as it does not involve an illegal operation (like dividing by zero, or taking the logarithm of a negative number). ...
- 3,178
1
vote
Method of Solving $5^x=326$ (Logs not allowed)
This may be a type of problem that is both funny and educational, provided that
a) You clearly state the rules. "Don't use ..." is way too vague and can result in all sorts of arguments. ...
- 4,439
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