I am having trouble figuring out how I can solve this log:

$$6 \log(x^2+1)-x=0$$

The steps i've thought to take so far are as follows:

step 1: subtract the right most x to the other side of the equation:

$$6 \log(x^2+1) = x$$

step 2: divide by 6:


step 3: make both sides an exponent of 10 to get rid of the log:

$$x^2+1 = 10^{\frac{x}{6}}$$

step 4. ??????????????

  • 1
    $\begingroup$ Note that $0$ is a solution. $\endgroup$ – Raskolnikov Mar 26 '11 at 22:21
  • $\begingroup$ The other solution is about $13.6267$. These are the only two real solutions. $\endgroup$ – Raskolnikov Mar 26 '11 at 22:29
  • $\begingroup$ @Raskolnikov: are you sure? 0.1690366 and 45.92765242537 look close to solutions $\endgroup$ – Henry Mar 26 '11 at 23:08
  • $\begingroup$ there are three solutions according to my textbook and wolframalpha if you wanna paste the equation over there. I just dont know How to get to the solutions $\endgroup$ – Matt Mar 26 '11 at 23:10
  • $\begingroup$ @Henry: Could you have used the natural logarithm instead of the logarithm base 10? @Matt: Should you actually use the natural logarithm instead of logarithm base 10? ;p $\endgroup$ – Raskolnikov Mar 26 '11 at 23:24

So $$x^2+1 = 10^{x/6}$$ or $$x^2-10^{x/6} +1 = 0$$ So one can use Newton's Method (i.e. choose the initial point to be $x = 0$).

  • 2
    $\begingroup$ I.e. there's no analytic solution, although you can find an approximate solution using various methods. Newton's method is a numerical method for finding approximate solutions. $\endgroup$ – Yuval Filmus Mar 26 '11 at 22:29
  • $\begingroup$ How would I solve for x after subtracting the 10^(x/6)? $\endgroup$ – Matt Mar 26 '11 at 23:10

To find the biggest root, you can also iterate the equation as follows:


Start for instance with $x_0=1$.

To find the other root besides the trivial $0$, you can iterate the following equation using the same seed as before:


Finally, you could show by an analysis of the function $6\log(x^2+1)-x$ that these are the only three possible solutions.

EDIT: Extra reference on fixed point iteration.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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