This equation looks hard but is really not. It is easy to solve with algebra, provided you already learned logarithm. But I challenge you to solve this by a style of ‘conversing in plain English.’
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I’m assuming that it’s cheating to just express the appropriate algebraic manipulation in words? You can sort of solve it by inspection and judicious trial-and-error.
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If you have a nice round number like ‘2’ dropping out of a log() function, that means you’ve taken the log of 1 followed by that number of zeros. In this case 100.
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So 98+(something complicated) = 100. The complicated bit is obviously equal to 2, which is the square root of 4.
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We might observe that if x were 3, then -12x+36 just disappears, but that leaves us with x2=9, which isn’t what we want. It’s close, though, so let’s experiment around that value on the assumption that x is a round number. By just increasing x to 4, we get 16-48+36 = 4.
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8, 4
Hmm. Yeah, it’s less obvious how to get the second solution by inspection.
- Wait till Finley comes up with the answer.
- Since he’s always half right double it and that’s the correct answer.
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I haven’t done any math even remotely related to this in over 30 years. It might as well be in a completely different language.
Difficult is relative.
Wouldn’t you just rewrite the bit inside the square root as (x − 6)2 such that the square root is canceled out, which gives 98 + x − 6 = 100 as you wrote, then x is obviously 8?
I don’t think it’s necessary to fiddle with numbers and trial and error here, though I realize you’re doing it as an engineer…
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I was trying to avoid any explicit algebra at all. Or at least the appearance of algebra.
But yeah, getting both solutions to a quadratic equation that way is not easy.
I’m not sure how you’re supposed to discuss √(x2 − 12x +36) without at least a bit of algebra…
It’s certainly more difficult than it looks when you can’t just plug 1, -12 and 32 into the standard formula.
Yes, absolutely.
That’s one way to approach the correct answer.
When I tried this question, I reason for this step in this way:
There are two numbers, one is 12 less than the other. And their product must be minus 32. One guess that one is 4 and the other is minus 8 happens to work.
The point of the challenge that I propose in this question is ‘don’t dive into algebra for solution when you see an equation with x’) . You don’t have to solve every x or y in every equation all the time.
I saw this question from a tweet. Almost everyone that responded used algebra to solve the question. That was not fun, just rote exercise.
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The question is literally asking for the solution to x though, and the equation contains algebra to begin with. Imposing the requirement that people shouldn’t use algebra to solve algebra, and instead fumble their way through by guessing and seeing what comes out, is silly.
It’s just trying to do algebra with extra steps.
I really don’t understand what you’re trying to say with the bolded part. What are the “two numbers” here - x2 and 12x, and why is it the case that one must be 12 less than the other? (Without using algebra, remember.)
I challenge you to fully explain this in plain English and derive the value of x as required by the question without using any algebra at all. 
Is that a natural log or base-10 log?
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Base 10 of course. It’s standard nomenclature (when there’s no subscript).
Not in most graduate-level textbooks.
Base-10 logs have virtually no use in science.
- Exponentiate both sides.
- Subtract 98 from both sides.
- Square both sides.
- Solve for x.
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That’s not true at all, including in graduate-level textbooks (and beyond). “log” to mean “log10” is completely standard in scientific publishing. Maybe not in mathematics, but in science definitely. Natural log would be “ln”.
It comes up pretty often as well.
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I actually agree with OysterOmelet, a lot of times natural log is implied/obvious and they just write log. Especially in graduate math and physics texts because you’re talking about them. I’ve only seen ln in high school books.
Of course in this case it’s obvious which one is used. He’s just asking for the sake of asking.
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Maybe it depends on the field then. 
I essentially only ever see “log” (for log10) or “ln” (for loge) in the academic papers I work with (mostly chemistry/materials, some physics and biology), and those versions are also mandated or accepted by at least a couple of major scientific publishing style guides.
I don’t think I’ve ever seen “lg” either, but I think some standards or whatever prefer that. I occasionally see “log10”, but never “loge”.
Not sure about really heavy math papers - I rarely see them.
Yeah, true.
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