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- Thread starter darthchocobo
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matt grime

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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).

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HallsofIvy

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[tex]log_{10}(A \cdot 10^n)}= log_{10}(A)+ n[/tex]

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)= e

[tex]\frac{de^x}{dx}= e^x[/tex]

From that

[tex]\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.

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arildno

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What is the use of having two ways to go to the supermarket???

- #5

HallsofIvy

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Darned if I know!

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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 [/tex]

The solution is [tex]x=\mbox{log}_{10} 53[/tex].

On your calculator you will find**log** and apply it to 53.

Similarly, for

[tex] e^x = 7 [/tex]

the solution is [tex]x=\mbox{log}_{e} 7[/tex]. 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

[tex] 2^x = 14 [/tex] you have to use [tex]\mbox{log}_{2}[/tex].

But how do you find x now if there is no key for [tex]\mbox{log}_{2}[/tex] on your

calculator?

Of course not all logarithms are on your calculator,

but you can calculate them by a formula (change of base):

[tex] \mbox{log}_{2} 14 = \frac{\mbox{log}_{10} 14}{\mbox{log}_{10} 2}[/tex]

Why did we use [tex]\mbox{log}_{10}[/tex] 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

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 [/tex]

The solution is [tex]x=\mbox{log}_{10} 53[/tex].

On your calculator you will find

Similarly, for

[tex] e^x = 7 [/tex]

the solution is [tex]x=\mbox{log}_{e} 7[/tex]. On your calculator

you'll have to press the

(ln stands for natural logarithm.)

There are also other logarithms, for example if you want to solve

[tex] 2^x = 14 [/tex] you have to use [tex]\mbox{log}_{2}[/tex].

But how do you find x now if there is no key for [tex]\mbox{log}_{2}[/tex] on your

calculator?

Of course not all logarithms are on your calculator,

but you can calculate them by a formula (change of base):

[tex] \mbox{log}_{2} 14 = \frac{\mbox{log}_{10} 14}{\mbox{log}_{10} 2}[/tex]

Why did we use [tex]\mbox{log}_{10}[/tex] on the right hand side of the equation?

Because this key is on the calculator! It's the

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

Last edited:

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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.

- #8

mathwonk

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always use natural log, never use common log.

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

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HallsofIvy

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always use natural log, never use common log.

Is that an order?:rofl: I'm remarkably bad at obeying orders!

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symbolipoint

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- #12

arildno

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These are retained for inertial reasons; i.e, people have gotten so used to the actual values appearing in the formula that they are loath to change them by making a logarithm shift.

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