Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?

share|cite|improve this question
Is the log a natural log? Or base 10? – ncmathsadist Oct 11 '12 at 1:17
@ncmathsadist Sorry, I should have specified. Its base 10. – abc Oct 11 '12 at 1:18
Are you looking for real solutions or complex solutions? – S.B. Oct 11 '12 at 1:19
@S.B. I am looking for all real solutions. – abc Oct 11 '12 at 1:20
@abc: It can have at most one real solution, because the LHS is strictly increasing while the RHS is strictly decreasing (actually, it has exactly one solution). Obviously you must look at $x>15$. I'm not sure if you can find a solution explicitly, but you can solve it numerically. – S.B. Oct 11 '12 at 1:24

Hint: Consider left and right sides at $x=16$ and $x=17$. You won't find an explicit "closed-form" solution, but you can prove that it exists.

share|cite|improve this answer
Thanks Robert. The solution is approximately 16 (which I found through just substituting the values in). Sorry, but my mathematical skills are relatively elementary - what is a "closed form solution"? The Wikipedia article on it was very rigorous and hard to understand. – abc Oct 11 '12 at 1:29
@abc: "closed form" solution means a formula using only elementary operations (addition, substraction, multiplication, division) plus a handful of functions (exponentiation, roots and logarithms, trigonometric). – Javier Oct 11 '12 at 1:54
"Closed-form" is a rather subjective term: basically you allow "well-known" functions, but people will differ on precisely which ones to include. – Robert Israel Oct 11 '12 at 4:59

Put \begin{equation*} f(x) = 3\log_{10}(x - 15) - \left(\dfrac{1}{4}\right)^x. \end{equation*} We have $f$ is a increasing function on $(15, +\infty)$. Another way, $f(16)>0 $ and $f(17)>0$. Therefore the given equation has only solution belongs to $(16,17)$.

share|cite|improve this answer
Thanks. What you did above seems to be just an approximation - isn't there a formal way to find an exact solution? – abc Oct 11 '12 at 1:50
Exact solution is too difficult to find. – minthao_2011 Oct 11 '12 at 2:17
At x=16 we have $log_{10}(16-15)=0$ so $f(16)<0$. I guess that's what you should have written in your answer anyway, since the statement $f(17)>0$ is right. You want a sign change to guarantee a zero in between. – coffeemath Oct 11 '12 at 2:25

Let $x=16+y$. After approximating $\log(1+y)$ with $y - \frac{y^2}{2}$, $(\frac{1}{4})^y = \exp(-\log(4) y)$ with $1 - \log(4) y$, $\frac{1}{1+\epsilon}$ with $1 - \epsilon$, get $$y \approx \frac{\log(10)}{3 \cdot 4^{16}}.$$

WIMS Function Calculator gives the exact solution, $1.7870412306 \cdot 10^{-10}$ compared to the approximation, $1.7870412309 \cdot 10^{-10}$.

share|cite|improve this answer
Please check if my LaTeX-editing did not mess up your equations. In any case, (+1) for the nice approach. – TMM Dec 31 '12 at 0:42

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.