# Formula for finding logarithms possible?

1. Sep 5, 2009

### Thymo

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...

Thanks in advance for any answers.
~ Thymo

2. Sep 5, 2009

### espen180

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.

Last edited: Sep 5, 2009
3. Sep 5, 2009

### kotreny

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.

4. Sep 10, 2009

### Thymo

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

5. Sep 10, 2009

### CRGreathouse

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.

6. Sep 10, 2009

### slider142

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.

7. Sep 10, 2009

### Thymo

: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?

8. Sep 10, 2009

### Staff: Mentor

9. Sep 10, 2009

### Thymo

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

10. Sep 10, 2009

### CRGreathouse

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.

11. Sep 12, 2009

### Mentallic

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:

12. Sep 12, 2009

### HallsofIvy

Staff Emeritus
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