# I am approximating $\ln x$ and $\log x$. How could I make these curves into a general equation?

Because I am waiting for my graphing calculator to ship, I need a quick-and-dirty way to calculate logarithms on a four-function calculator (for when I need to keep my laptop away from where I work).

By modifying the common $n^{th}$ root on a four-function calculator trick, I have found the following curves easy-to-remember and fairly accurate (2-3 significant figures).

$$\ln x \approx 2^{10}(x^{\frac{1}{2^{10}}}-1)$$

$$\log x \approx 444(x^{\frac{1}{2^{10}}}-1)$$

Now, I would prefer something along the lines of

$$\log_bx \approx k(x^{\frac{1}{2^{10}}}-1)$$

where $k$ is something that relates to the base $b$ of the logarithm.

I realize that this may not be possible, or will be so unintuitive that it would be easier to calculate two natural logarithms and make use of that memory key. But hey, curiosity is the lust of the mind.

Edit: Just realized that it's not possible (second edit: without more logarithms), as setting up the equations

$$10^{k_1}=444$$ $$e^{k_1}=1024$$

just yield parallel lines. Any other suggestions on how to make $k$ for any base easy to solve for?

• Interesting but : how will you compute $x^{\frac{1}{2^{10}}}$ ? – Claude Leibovici Mar 23 '16 at 5:02
• Ten square roots. It's just ten square roots is a bit of a mess and hard to read. – Hello Mar 23 '16 at 5:29
• You can get four significant figures from a one-page table of logarithms: abitofauldmaths.org/2013/08/how-to-use-log-tables – David K Mar 23 '16 at 5:44
• I was initially against the use of a lookup table because I often slip up and use the value to the right of what I actually want. However, I realize this approximation is dumb and doesn't save any effort. Long nights. :p – Hello Mar 23 '16 at 14:33

You can just set $k=2^{10}/\ln(b)$ since $\ln(x)/\ln(b)=\log_b(x)$