# Formula for finding logarithms possible?

Is it possible to make a formula for logarithms of any base?

logb(a)=x

I want to find x through some formula. I've seen that you can use a series for e as the base, is that the only base that can be solved for?

Is there any work being done to accomplish this, or, maybe it has been proven impossible? In that case, how was it proven impossible?

Sorry if this is a noobish question... In my defence, a noob can't recognize a noobish question...

~ Thymo

## Answers and Replies

Sure it's possible. If you know one base, you know all bases.

The equation you are looking for, is: $$\log_k x=\frac{\log_n x}{\log_n k}$$ where k and n can be any positive number above 1. (Do you see why this formula works?)

So if you use e as your end base, you have a fraction bewteen two logarithms which equals to x.

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I don't suppose you're looking for the change of base formula, used when a calculator only has a log10( ) key:

logb(a) = log10(a)/log10(b)

Actually you don't have to use base 10; anything, including e, is workable. (Of course, you can't use 1, 0, negative numbers, etc.)

If this looks new I think I can derive it, free of charge.

Everyone is a rehabilitated noob.

But this doesn't really help me do logs without a calculator does it? ...

CRGreathouse
Homework Helper
But this doesn't really help me do logs without a calculator does it? ...

If you want to do it by hand you should use a table/slide rule. Failing those you're going to have to use Newton's method or something similar.

Is there any work being done to accomplish this, or, maybe it has been proven impossible? In that case, how was it proven impossible?

The logarithm is a transcendental function. The only way to reduce it to something like a polynomial with rational coefficients is through infinite series like Taylor's or iterative processes like Newton-Raphson. However, practical applications use either logarithmic tables/slide rules or calculating machines.

The logarithm is a transcendental function. The only way to reduce it to something like a polynomial with rational coefficients is through infinite series like Taylor's or iterative processes like Newton-Raphson. However, practical applications use either logarithmic tables/slide rules or calculating machines.

:grumpy:

So it's possible to use either a Taylor Series or "Newton-Raphson"-method(??). There EXISTS a formula that makes it possible, but impractical? Does it EXIST? Any LINKS or EXPLENATIONS?

Borek
Mentor
Thanks! It seems as if I have a lot to work with... At least it's something... ;)

CRGreathouse
Homework Helper
There EXISTS a formula that makes it possible, but impractical?

You already know it's possible, if you accept that a calculator can do it. You could simply make a circuit diagram of the calculator and trace through its operation when given the appropriate set of keystrokes. That's far less practical, but possible.

Mentallic
Homework Helper
You already know it's possible, if you accept that a calculator can do it. You could simply make a circuit diagram of the calculator and trace through its operation when given the appropriate set of keystrokes. That's far less practical, but possible.

That might not necessarily mean a formula exists. The calculator could just be estimating values for the exponential and then by much trial and error, give a 10 decimal value display :tongue2: