Solving logarithmic equations

Posted on by | 5 Replies

Selected by: Christopher Danielson

Question: What’s up with this? Why does this piece of student work matter? Does it?

  • One thing I notice is that the student isn’t exactly solving for x — they aren’t “undoing” or using inverse operations. Solving for x might look more like:

    log[2] y = log[2] 2^x
    log[2] y = x * log[2] 2
    log[2] y = 1x

    Another thing I notice is that they’re applying a rule they’ve learned: y = x^z -> log[x] y = z. I wonder if there are questions they will struggle with because their understanding is relatively procedural.

    I wonder how they would solve a problem like find log[x] x^3, or, more weirdly, find z if log[x] z = x^3.

    I wonder how they would solve a problem like if log[2] 5 = a, and log[2] 4 = b, find log[2] 20.

    I’m not sure what other problems would diagnose more about this student’s conceptual understanding of logs. Help, team!

  • l hodge

    It seems like they want to see the problem (y = 2^x) in the form of a template: y = x^z. Usually the template is taught as y = b^x – probably they remembered the form incorrectly. In any event, they substituted into the template (replace 2 with x and x with z), probably did the work Max suggests, but didn’t substitute back to the original variables when done (switch z back to x).

    The work looks more like taking log[2] of both sides (using inverse operations) & simplifying the right side all in one step, but the switch to z makes me think maybe they are using a verstion of the rule Max mentioned. Most students I see that use the rule (definition) Max referred to would end up with y on the other side of the equal sign because the definition is usually written: log[b] (y) = x iff b^x = y.

    I am with you, Max, in preferring to apply inverse operations to both sides of the equation. But, applying inverse operations to each side is a rule to memorize as well.

    • Jim Doherty

      I get what you’re saying about the application of inverse functions being a rule to remember, but it is FAR more than that. It is a habit to adopt and one to apply to almost any equation that you encounter. If you conceptually link the idea that exponents and logs are inverses to the fact that multiplying/dividing are inverses and adding/subtracting are inverses and cosine and arccosine are inverses, etc. then you’ve made a breakthrough in working with your students and they will have made a breakthrough in untangling equations.

      • l hodge

        agreed

  • I also see this as matching to a template rather than solving a problem.

    To me, there are equations to solve with logs, like 5 * 2^x = 8, where I would definitely think in terms of inverse operations, but then there are equations like the one given here, which I think of more in terms of “fact families” or definition of logs, rather than inverse operations.

    I tend to teach that 2 * 3 = 6 and 6 / 3 = 2 and 6 / 2 = 3 and so on are all different ways of writing the same fact, and I encourage people to rewrite among those by thinking of it that way, not only as “divide both sides by 3” or the like. Similarly, I teach that 2^3 = 8 and cube root of 8 = 2 and log base 2 of 8 = 3 are different ways of writing the same fact (with some caveats about negative numbers, similar to the needed caveats about 0 in the division case). So my students might see y = 2^x and x = log base 2 of y as synonyms, more than as the result of a process of taking the log base 2 of both sides and using the fact that log base 2 and exponents to base 2 are inverses.

    When my students see a log statement they don’t understand, their habit would be to take the log part — say, log base x of 42 or something — and introduce a variable, say t, to equal it, and then write the synonym, x^t = 42, to develop their understanding of what t means.

    I feel like there’s something there that kids don’t get from the “log base x is the inverse of ‘x to the'” kind of statements, but at the same time I feel like there’s something missing in the way that they then end up seeing logs (and especially equations involving them, and the process of solving them) as a different sort of beast than the other things they learn about in an algebra 2 class.