# Point of logarithms?

Why do you need logarithms? In what situations do you use them?

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When the numbers are too big or too small. For instance, pH is a more "manageable" number than the corresponding concentration of hydronium ions in an acid. –  non-expert Jan 4 '11 at 18:19
Every branch of mathematics and science uses logarithms. There is a list at the Wikipedia article: en.wikipedia.org/wiki/Logarithm#Uses_and_occurrences . Can you ask a more precise question? –  Qiaochu Yuan Jan 4 '11 at 18:26
I am voting to close as not a real question. –  Aryabhata Jan 4 '11 at 18:56
It would be more interesting to answer if you picked certain details of what you want to know. What do you mean by "the point"? What do you already know about logarithms? What else do you want to know or do not understand? –  Pedro Tamaroff May 1 '12 at 15:23
Possible duplicate: math.stackexchange.com/questions/35810/… –  user26649 May 1 '12 at 15:35
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Logarithms are primarily used for two thing:

i) Representation of large numbers. For example pH(the number of hydrogen atoms present) is too large (up to 10 digits). To allow easier representation of these numbers, logarithms are used.

For example let's say the pH of the substance is $10000000000$. This can written as $10^{10}$.

Or let the pH of another substance be $1000000$. This can be written as $10^6$. Note the base is always the same, but the exponent is unique. Therefore the log of the substance can be used to identify the substance. For example the first substance can be represented as $log$ $10000000000$ or $10$ and the second substance can be represented as $log$ $1000000$ or $6$. Note $6$ and $10$ are much easier to deal with.

But what if you're not a chemist? How would you use logs?

ii) Algebra.

Let's say you have the equation $316 = 10^x$. How would you solve for $x$? You could find the log of $316$ which is approximately $2.5$. The equation would then be $10^{2.5} = 10^x$. Therefore $x$ is $2.5$. Logs are therefore extremely useful when solving for exponents.

Note that although I have restricted my examples to log base 10 for simplicity, logs can exist in other bases. For example $\log_2 32$ (log to the base 2 of 32) is $5$ since $2^5= 32$. Other important log bases include the the natural log, which is commonly used in advanced mathematics.

What other appliances do logs have? Logs have a variety of real life applications such as calculating half lives and exponential growth/decay. In fact the inverse of an exponential function is a logarithmic function!

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You can "undo" addition by performing subtraction. You can "undo" multiplication by performing division.

When it comes to exponents, $x^y \not= y^x$, so you need two different "undo" functions.

Suppose that you know the value of $v$, and you know that this value was calculated by $v = x^n$.

• If you know what $n$ was, then you can "take the $n$th root of $v$" to find $x$. That is, $x = \sqrt[n] v$.

• If you know what $x$ was, then you can "take the base-$x$ logarithm of $v$" to find out what $n$ was. That is, $n = \log_x v$.

So that's what the logarithm function does. Why is that useful? Well, for the same reason that being able to undo an addition or a multiplication is useful. It lets you work backwards through a calculation. It lets you undo exponential effects.

Beyond just being an inverse operation, logarithms have a few specific properties that are quite useful in their own right:

• Logarithms are a convenient way to express large numbers. (The base-10 logarithm of a number is roughly the number of digits in that number, for example.)

• Slide rules work because adding and subtracting logarithms is equivalent to multiplication and division. (This benefit is slightly less important today.)

• Lots of things "decay logarithmically". For example, hot objects cool down, cold objects warm up. Things in motion experience friction and drag and gradually slow down.

• If you can take a problem and split it into two smaller problems that can be solved independently, you can probably write a computer program where the number of steps required to solve the problem is "logarithmic". That is, the time taken depends on the logarithm of the amount of data to be processed.

I'm sure there are lots of other examples.

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Logarithms are defined as the solutions to exponential equations and so are practically useful in any situation where one needs to solve such equations (such as finding how long it will take for a population to double or for a bank balance to reach a given value with compound interest).

Historically, they were also useful because of the fact that the logarithm of a product is the sum of the logarithms and sums are easier to calculate by hand (or to estimate by overlapping rulers as in a slide rule). In addition to providing a computational "trick", this property is the basis of the mapping property described in Christian Blatter's answer and generalizes to the concept of self-adjoint generators of unitary groups, which has many mathematical applications and relates physical observables to symmetry properties in quantum mechanics.

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BTW, it can be proved that if a function $\varphi$ satisfies $\varphi(xy)=\varphi(x)+\varphi(y)$ then it must be that $\varphi(x)=c\log(x)$ for some constant $c$. –  Amihai Zivan May 18 '12 at 12:34
In the geometric view of real numbers there are two basic forms of "movements", namely (a) shifts: each point $x\in{\mathbb R}$ is shifted a given amount $a$ to the right and (b) scalings: all distances between points are enlarged by the same factor $b>0$. In some instances (e.g. sizes of adults) the first notion is appropriate for comparison of different sizes, in other instances (e.g. distances between various celestial objects) the second notion. The logarithm provides a natural means to transform one view into the other: The sum of two shifts corresponds to the composition of two scalings.