Logarithm Calculation with Windows 7 Calculator

Question

I would like to use the Windows Calculator in Scientific Mode in order solve a very basic Logarithm equation but, unfortunately, I couldn't do that.

Here is the problem:

log_5 125=?

Thank you very much for your help...

Well, I know it equals to "3", but, how can I use the Windows Calculator to get computed that equation for example?

Answer 1

You can calculate a logarithm in a given base by calculating two logarithms in an arbitrary base, using the following equation:

log_b (x) = log_k (x) / log_k (b)

As the windows calculator got a ln button, which stands for the natural logarithm (that is, log in basis e,) then you can press 125, ln, /, 5, ln, and get the desired result.

For bonus points, here is why the above equation holds:

• Let a^b = c. Remember that this sets b = log_a (c).
• Take log_k of both sides of the first equation. We get: log_k (a^b) = log_k (c)
• Using the logarithmic identity log (x^y) = y * log (x), we get b * log_k (a) = log_k (c)
• Hence b = log_k (c) / log_k (a).
• From the first step, we have b = log_a (c), hence log_a (c) = log_k (c) / log_k (a). QED.