1

Solve for $x$: $3 \log_{10}(x-15) = \left(\frac{1}{4}\right)^x$

I seem to get stuck when I get to logarithm of a logarithm or power of a power, graphing it and doing some guess and check on the calculator shows that $x$ should be just a bit above 16, but I would like to know how to figure it out algebraically if possible. I'm in Grade 11 so I probably won't understand anything too complicated.

Intradiction
  • 77
  • 6
  • Is $\log$ base 10 or natural? – Andrew Li Jun 06 '18 at 21:53
  • 10, sorry I thought log with no base defaults to 10 always – Intradiction Jun 06 '18 at 21:55
  • It's common in higher level math for $\log$ to mean the natural logarithm. So typically one writes $\log_{10}$ for the base-10 logarithm to be unambiguous. – arkeet Jun 06 '18 at 21:57
  • OK, thanks I will remember that. But what's the point of that convention if there is ln(x) to mean natural logarithm? – Intradiction Jun 06 '18 at 22:04
  • Ooog. Don't get us started. Mathematicians are a prickly bunch. That's the point of the convention. ... Okay, mathematicians don't believe $10$ has any significance at all and the only log that does is the natural log and because natural log is the norm, not the exception, to have a terminology $\ln$ is perverse and unnecessary.... Look, just ... let it go.... – fleablood Jun 07 '18 at 06:12
  • Where is this question coming from ? If it is a Grade 11 exercise, then it is possibly mis-copied. –  Jun 07 '18 at 06:35
  • Thanks for the responses guys. Also this is not mis-copied, I asked my teacher and she also agreed that it was an unfair question for us. – Intradiction Jun 08 '18 at 18:11

2 Answers2

2

Consider that you look for the zero of $$f(x)=3 \log_{10}(x-15) - \left(\frac{1}{4}\right)^x=\frac{3 }{\log (10)}\log (x-15)-4^{-x}$$ $$f'x)=\frac{3}{(x-15) \log (10)}+4^{-x} \log (4)$$ Since you notice that $x$ should be just a bit above $16$, perform one single iteration of Newton method writing $$0=f(16)+f'(16)(x-16)\implies x=16-\frac{f(16)}{f'(16)}$$ This should give $$x=16+\frac{1}{\log (4)+\frac{12884901888}{\log (10)}}\approx 16.000000000178704123046$$ while the "exact" solution would be $16.000000000178704123062$

Claude Leibovici
  • 260,315
1

Note that

  • $f(x)=3\log_{10} (x-15)$, defined for $x>15$, is strictly increasing

  • $g(x)=\frac1{4^x}$ is strictly decreasing

and

  • $f(16)=0<g(16)$

  • $f(25)=3>g(25)$

then by IVT a solution exists for $x\in(16,25)$ which can be found by numerical methods.

user
  • 154,566
  • What does IVT mean? Also I understand that there exists a solution for x, but what exactly is the "numerical method" for finding it? – Intradiction Jun 06 '18 at 22:01
  • @Insanit https://en.wikipedia.org/wiki/Intermediate_value_theorem – user Jun 06 '18 at 22:03
  • ok I get that there is a solution but could you please show step by step for how to algebraically solve this equation? – Intradiction Jun 06 '18 at 22:06
  • @Insanit We can't find explicit derivation bt elementary functions, we need numerical methods to obtain the solution. – user Jun 06 '18 at 22:12
  • I'm not sure what that means. Could you please rephrase that in layman's terms? Remember that I am in Grade 11, I'm sorry if it's frustrating to explain to me :( – Intradiction Jun 06 '18 at 22:14
  • @John so you mean you have to get a computer to solve it? Is the answer an irrational number then? – Intradiction Jun 06 '18 at 22:22
  • Sorry, I deleted my comment because I thought it was an integer, but it isn't. Yes, that's basically what I'm saying. The answer is a little bit more than $16$: http://www.wolframalpha.com/input/?i=3+log+(x-15)+%3D+(1%2F4)%5Ex&rawformassumption=%7B%22FunClash%22,+%22log%22%7D+-%3E+%7B%22Log10%22%7D – John Jun 06 '18 at 22:23
  • @John OK, thanks. – Intradiction Jun 06 '18 at 22:27
  • @ClaudeLeibovici Ops! Thanks Claude, I fix it! – user Jun 07 '18 at 05:00
  • "could you please show step by step for how to algebraically solve this equation?" The thing is you can't. Just like there is no algebraic way to calculate some arbitrary sine. It doesn't actually mean you need a computer to solve or that it is irrational (although it almost certainly is). But the only thing we can do is either a) approximate until we get tired. b) since we know it exist, say "we will call the number George and treat it as a known constant like $\pi$ or $e$ or any other number that can be described, approximated, but not expressed. – fleablood Jun 07 '18 at 06:21