Finding the logarithm of a number using a hanging chain: Leibniz’s definition of catenary curves!

We know, how simple it is to multiply two numbers written in exponents forms (when same base). We just add the power.?

3^5 *3^7 = 3^12.?

Can we do the similar working to multiply 312* 517.?

Of course we can, but we need to express 312 and 517 both in exponent form of?same base. Logarithms is the key to do this.

Using?log?we can express any number in terms of powers of any other number (provided these numbers satisfy some conditions).

?How many?2s do we multiply to get?8?

Answer:?2 × 2 × 2 = 8, so we needed to multiply?3?of the?2s to get?8

So the logarithm is 3

How to Write it

We write "the number of 2s we need to multiply to get 8 is 3" as:

log2(8) = 3

?So these two things are the same:

The number we are multiplying is called the "base", so we can say:

·???"the logarithm of 8 with base 2 is 3"

·???or "log base 2 of 8 is 3"

·???or "the base-2 log of 8 is 3"

Notice we are dealing with three numbers:

·???????the?base: the number we are multiplying (a "2" in the example above)

·???????how many times to use it in a multiplication (3 times, which is the?logarithm)

·???????The number we want to get (an "8")

More Examples

Example: What is?log5(625) ?

We are asking "how many 5s need to be multiplied together to get 625?"

5 × 5 × 5 × 5 = 625, so we need?4?of the 5s

Answer:?log5(625) = 4

Example: What is?log2(64)?

We are asking "how many 2s need to be multiplied together to get 64?"

2 × 2 × 2 × 2 × 2 × 2 = 64, so we need?6?of the 2s

Answer:?log2(64) = 6

Exponents

Exponents and Logarithms are related, let's find out how ...

The?exponent?says?how many times?to use the number in a multiplication.

In this example:?2^3?= 2 × 2 × 2 = 8

(2 is used 3 times in a multiplication to get 8)

So a logarithm answers a question like this:

In this way:

The logarithm tells us what the exponent is!

In that example the "base" is 2 and the "exponent" is 3:?

So the logarithm answers the question:?

What exponent do we need?

(for one number to become another number)??

The?general?case is:

Example: What is?log10(100) ?

102?= 100

So an exponent of?2?is needed to make 10 into 100, and:

log10(100) = 2

Example: What is?log3(81)?... ?

34?= 81

So an exponent of?4?is needed to make 3 into 81, and:

log3(81) = 4

?Common Logarithms: Base 10

It is how many times we need to use 10 in a multiplication, to get our desired number.

Example:?log(1000) =?log10(1000) = 3

?Natural Logarithms: Base "e"

Properties of Logarithms:

Interestingly, finding value of logarithms of any number is not that tough.?Leibniz?definition of?catenary curves?can be used smartly to find the log values.

Here are the steps,

?Step (1):?Suspend a chain from two horizontally aligned nails. Draw the horizontal through the endpoints, and the vertical axis through the lowest point.

Step (2):?Put a third nail through the lowest point and extend one half of the catenary horizontally.

Step (3):?Connect the endpoint to the midpoint of the drawn horizontal, and bisect the line segment. Drop the perpendicular through this point, draw the horizontal axis through the point where the perpendicular intersects the vertical axis, and take the distance from the origin of the coordinate system to the lowest point of the catenary to be the unit length. We will show below that the catenary now has the equation y = (e? + e?? )/2 in this coordinate system.

Step (4):?To find log(Y ), find (Y + 1/Y )/2 on the y-axis and measure the corresponding x-value (on the catenary returned to its original form). This assumes that Y > 1. To find logarithms of negative values, use the fact that log(1/Y ) = ? log(Y ). If you seek the logarithm of a very large value, then you may end up too high on

the y-axis; in such cases you can either try hanging the endpoints closer together or using logarithm laws to express the desired logarithm in terms of those of lower values.

Happy Intuitive Learning!

Reference: Viktor Bl?sj?, "How to Find the Logarithm of Any Number Using Nothing But a Piece of String", The College Mathematics Journal, Vol. 47, no. 2, March 2016, pp. 95-100.

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