# Log(x), an easy and useful way to calculate it

• B
• guifb99
In summary, the conversation discusses an easy and useful way to calculate the logarithm of any natural number, including primes. The equation used is ½Logb(x2-1)≈logb(x), which is an approximation based on the property that logc(a*b)=logc(a) + logc(b). The conversation also mentions that this method may not be precise for certain values of k and n. Additionally, it is suggested to learn how to calculate logarithms by hand using Euler's method using 4 properties of logarithms. A link is provided for further information on this method.
guifb99
½Logb(x2-1)≈logb(x)

This is an easy and useful way to calculate the log of any natural number, including primes, it won't ever give a precise result, obviously (because of the -1), but as "x2-1" will always have divisors smaller than "x", you can easily calculate the approximation by using the property that logc(a*b)=logc(a) + logc(b).

It could actually just be n-1logb(xn-k)≈logb(x) but it would be harder to calculate depending on which "k" or which "n" you use and won't be as useful for school purposes.

Obviously, since ½logb(x2)=logb(x), the closer "k" is to 0, the more precise the result will be. And the closer "n" is to infinity, also the more precise the result will be, since the value of "k" will become less and less significant as "xn" gets bigger.

*It's good to notice that for a "k" bigger than one, it will fail miserably.

Hello guif,

How would you actually go about to calculate such a logarithm ? Can you give an example ? What if e.g. x = 23 ?

The general idea is the approximation ##f(x) \approx \frac{f(x-1) + f(x+1)}{2} ##.

for ##x > 1## , ##\log_b(x^2 -1) = \log_b((x-1)(x+1)) = \log_b(x-1) + \log_b(x+1)##

So instead of calculating ##\log 23## I look up ##\log 22 ## and ##\log 24## and then average ?

mfb
You still have to look up logarithms with your method. Basically if you have to look up two logarithms to get a third, you could have saved yourself work by simply just looking up the log you wanted first off. You are not saving anything, really.

Euler developed a simple way by hand using 4 properties of properties of logarithms. So learn how to do logs by hand, I did, it's fun.

dkotschessaa
jim mcnamara said:
You still have to look up logarithms with your method. Basically if you have to look up two logarithms to get a third, you could have saved yourself work by simply just looking up the log you wanted first off. You are not saving anything, really.

Euler developed a simple way by hand using 4 properties of properties of logarithms. So learn how to do logs by hand, I did, it's fun.:
https://www.fiziko.bureau42.com/teaching_tidbits/manual_logarithms.pdf

-Dave K
(sorry)

Last edited by a moderator:
I'm fixing it as we post !

#### Attachments

• manual_logarithms.pdf
116 KB · Views: 387
• manual_logarithms.pdf
116 KB · Views: 818

guifb99 said:
½Logb(x2-1)≈logb(x)
For large x, ##x^2 - 1 \approx x^2##, so ##\log_b(x^2 - 1) \approx \log_b(x^2) = 2 \log_b(x)##. Your equation above comes immediately from this one.

For example, with x = 25, ##\frac 1 2 \log(25^2 - 1) \approx 1.397592## and ##\log(25) \approx 1.39794##.

Last edited:
jim mcnamara said:
I'm fixing it as we post !

The one about square roots by hand is very cool also. We are never taught this stuff.

## 1. What is Log(x)?

Log(x) is a mathematical function that represents the logarithm of a number x. It is the inverse of the exponential function and is used to solve equations involving exponents.

## 2. Why is Log(x) useful?

Log(x) is useful because it can simplify complex mathematical expressions involving exponents, making them easier to solve. It is also used in many real-life applications, such as in finance and science, to model data and analyze trends.

## 3. How do you calculate Log(x)?

To calculate Log(x), you can use a calculator or a logarithm table. You can also use the natural logarithm function, ln(x), on a scientific calculator by entering "ln(x)" or "log(x)" and the number x. Additionally, there are many online calculators and apps that can calculate Log(x) for you.

## 4. What is the base of Log(x)?

The base of Log(x) can vary, but the most commonly used base is 10. This is known as the common logarithm. Another common base is e, which is the base of the natural logarithm function, ln(x).

## 5. What is the difference between Log(x) and ln(x)?

The main difference between Log(x) and ln(x) is the base used in the logarithm function. Log(x) has a base of 10, while ln(x) has a base of e. Additionally, Log(x) is used more frequently in practical applications, while ln(x) is used more in theoretical mathematics.

Replies
3
Views
2K
Replies
4
Views
1K
Replies
17
Views
2K
Replies
3
Views
933
Replies
5
Views
700
Replies
8
Views
3K
Replies
4
Views
3K
Replies
2
Views
2K
Replies
3
Views
1K
Replies
9
Views
3K