# Natural log or common log

## Main Question or Discussion Point

Can someone explain to me when to use natural log or common log? I understand that natural log gives creates a base e and the common base 10 but i don't understand why there are 2 different ones.

matt grime
Homework Helper
There aren't 2 bases for logs. There are infinitely many. You choose the base that is most appropriate for the question in hand. This will normally be base 2 or base e (very rarely base 10, you'll find later).

e is most useful because it satisfies f'(x)=f(x) if you know about differentiation, and comes up naturally (hint) in the solutions of the equations that cover most of real life.

Note that log(x) (log will mean base e unless other wise stated) and log_10(x) differ just by a multiplicative constant indendent of x. And in general logs to base b_1 and b_2 differ by a multiplicative constant depending only on b_1 and b_2.

The most obvious example of this is log_2(x)=2log_4(x), and notice that log_2(4)=2. In general the constant will be log_{b_1}(b_2).

HallsofIvy
Homework Helper
Base 10 was used extensively, at one time, because our number system is base 10 and it was easy to make tables of the logarithms of numbers between 1 and 2 and then use the fact that
$$log_{10}(A \cdot 10^n)}= log_{10}(A)+ n$$
Since logarithms could "convert" multiplication to addition, these were useful for doing calculations.

Now that calculators can quickly give logarithms of any numbers, or do your multiplications for you, "common" logarithms, base 10, are no longer so useful.

The natural logarithm, base e, is, of course, the inverse function to f(x)= ex which itself has the property that
$$\frac{de^x}{dx}= e^x$$
From that
$$\frac{d ln(x)}{dx}= \frac{1}{x}[/itex] a very useful (and even more useful integral). Common logarithms, base 10, are seldom used any more while natural logarithms, base e, are used a lot in calculus and higher mathematics. ______________________ Opinions are mine, and probably not those of any sane person. arildno Science Advisor Homework Helper Gold Member Dearly Missed What is the use of having two ways to go to the supermarket??? HallsofIvy Science Advisor Homework Helper Darned if I know! Hello darthchocobo, welcome to physicsforums, which log you have to use depends on the problem you want to solve. Suppose somebody asks you to solve the following: Find a number x for the following equation: [tex] 10^x = 53$$
The solution is $$x=\mbox{log}_{10} 53$$.
On your calculator you will find log and apply it to 53.

Similarly, for
$$e^x = 7$$
the solution is $$x=\mbox{log}_{e} 7$$. On your calculator
you'll have to press the ln key and apply it to 7.
(ln stands for natural logarithm.)

There are also other logarithms, for example if you want to solve
$$2^x = 14$$ you have to use $$\mbox{log}_{2}$$.
But how do you find x now if there is no key for $$\mbox{log}_{2}$$ on your
calculator?

Of course not all logarithms are on your calculator,
but you can calculate them by a formula (change of base):

$$\mbox{log}_{2} 14 = \frac{\mbox{log}_{10} 14}{\mbox{log}_{10} 2}$$

Why did we use $$\mbox{log}_{10}$$ on the right hand side of the equation?
Because this key is on the calculator! It's the log key in our first example from above.

See here for base change of logarithm:
http://hyperphysics.phy-astr.gsu.edu/hbase/logm.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/logm.html#c3

More on logs:
http://hyperphysics.phy-astr.gsu.edu/hbase/log.html

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if your in precalc or algebra II, you will probably be asked to evaluate logarithms using different bases (u actually use the common log or natural log to solve those). but afterwards, you dont have to worry about bases other than natural or common. In calculus (atleast for me), the only type of log used is the natural log. In chemistry, weve used natural and common logs. In physics, there was one occasion where natural logs were used. Basically, you will only be using natural or common logs.

As to distingush which to use, well, if there is ever e in a problem u will most likely be using the natural log, if the problem even calls for the use of a logarithm.

mathwonk
Homework Helper
always use natural log, never use common log.

D H
Staff Emeritus
Basically, you will only be using natural or common logs.
Not quite true. Base 2 is the "natural" base for logarithms in much of information theory and the theory of algorithms.

HallsofIvy
Homework Helper
always use natural log, never use common log.
Is that an order?:rofl: I'm remarkably bad at obeying orders!

symbolipoint
Homework Helper
Gold Member
Logarithms to base 10 will often be convenient. Most people are comfortable with base-10 numbers so graphs using logarithms to base 10 will seem more comfortable. The choice of base depends on what the analyst needs from the data. In case of any possible misunderstandings, just document clearly what was done. Analytical chemical/acid base equilibrium calculations, aside from physical studies, are usually done according to base 10; on the other hand, many redox calculations, relying on the Nernst equation are handled using base e (natural logarithm)

arildno