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From the perspective of mathematics education, I must ask: Why do you want to encourage students to solve such a problem without logarithms? There is a connection between square roots and logarithms (e.g., here) but I consider the mathematics that underlies it quite opaque without understanding $x\mapsto \log(x)$ beforehand. Personally, my inclination is ...

10

First, I'll answer the question posed by Benjamin Dickman: Solving problems with limitations is good practice for working with algebraic structures that do not have analogous functions. For example, solving $5^x \equiv 326 \mod 331$ is a situation where the log button on your calculator isn't going to help at all. (Neither is the bisection method.) So you ...

9

One could solve this by "guess and check", together with the knowledge that $5^x$ is increasing. So start with your observation that $3 < x < 4$. Test $3.1, 3.2, 3.3, ...$ and observe that $3.5<x<3.6$. Repeat the process for the next two decimal places. I am not sure if this is what they intended. Another method which could be used if they ...

9

I did this with my students a while ago. First I got them to construct their own slide rules using Briggs estimation technique. Then a slightly more accurate table translating between base 10 logs, decibels, base 2 logs, and musical notes ($semitone^{12}=2$). After that we studied Briggs' methods to improve the estimates and create accurate logs. My source ...

8

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 (elementary to college level), students who think of "moving" terms around can almost never generalize to the next level.

6

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 inverse function means! Misunderstanding/forgetting the "one input $\to$ one output" defining feature of a function. This issue arises because implicitly-...

6

This method does not give 3 digits precision, so is not really an answer to the original question -- but if all one is looking for is a quick-and-dirty approximation, here is a strategy that is very elementary. First of all, let's change the problem to one that will be easier to solve: $5^x=325$. Since we are only looking for a reasonable approximation to ...

5

Gauss said "You have no idea how much poetry there is in a table of logarithms." The first paragraph of this paper might get you pointed in the right direction ON THE DISTRIBUTION OF PRIMES—GAUSS’ TABLES

4

This problem is not hard with Briggs method. It is similar to the method described by Benjamin Dickman but converges with slightly less square roots. Take repeated square roots until the number is close enough to 1 to enable linear interpolation of the exponential. Do not take the same number of square roots for the other number. Instead take as many square ...

4

This is my method of solving. It uses the square function, and division. One then must accumulate the powers of 1/2, i.e 1/2,1/4,1/8, etc that are noted. Method - A) Divide by the base, in this case 5, and determine the exponent has a 3 prior to the decimal. B) Take the result and square it. C) If you can divide the result by 5, a binary 1 is noted. ...

4

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 $\frac{1}{2} < 0$ or something similar? I find students will be unsure rather than actively wrong about the relationship between $\frac{1}{2}$ and $0$. But it's very easy ...

4

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 collect data on your students' possible answers by giving them a fill-in-the-blank quiz on inverse functions. Then, keep track of the most-common wrong answers by ...

3

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 system B, and the inverse converts in the opposite direction, from B to A. There are circumstances where both are needed.           &...

2

There is only one way (that I know of) to solve this without knowledge of Calculus (and a calculator--that is without tables--although my solution will rely on the computational power of a calculator). First, I do not see the question as appropriate for the level of algebra or trigonometry (I certainly don't see the link to trigonometry). Frankly, I don't ...

1

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). But there are an infinite number of choices of "things" you can do to both sides, most of which don't help solve the problem. The student needs to know what the ...

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