Math Help

Intuitively, characteristic is saying about how big is the number on an exponential scale. If you remember your scientific notation, the -2 in log(0.0624) = -2 + 0.7952 is exactly the same thing as 0.0624 = 6.24 * 10^-2. So the idea is, in the grand scheme of every number in the world, 0.0624 is somewhere between 0.1 (characteristic = -1) and 0.01 (characteristic = -2).
Mantissa is saying how close/far away is the number from either end of the bracket - let's ignore for now.
The point about looking at characteristics, and why do we need to know logs in the first place, is that we are freakin' good at analyzing linear relations (you know, slope, intercepts, those kind of stuff), but most things in nature behaves exponentially. Logarithms turn exponential relations into linear relations.
Anyhow, that's a generally approachable first introduction to logs. For maths folks though, we just know log by definition.
You might want to watch this video by 3blue1brown on logarithms.

Hey guys, I was doing some pre reading and came across this part under the logarithms chapter. What’s the point of looking for the mantissa and characteristics if it’s still pretty far off from its true value? Or are we finding the characteristics and mantissa as a way to estimate the values of common logs?
For example, log(0.0624) = -2 + 0.7952 (characteristic and mantissa) BUT if you put it into a calculator it’s -1.2048

Let's break down your question and analyze the concepts step by step:
1. Definition of Characteristic and Mantissa:
- The characteristic is the integer part of a logarithm.
- The mantissa is the fractional part of a logarithm.
2. Purpose of Characteristic and Mantissa:
Before the widespread use of calculators and computers, people used logarithm tables for complex multiplication and division. The concepts of characteristic and mantissa made this process more manageable.
3. Your Example: log(0.0624) = -2 + 0.7952
This expression is actually not correct. Let's look at the proper calculation:
log(0.0624) = log(6.24 × 10^-2)
= log(6.24) + log(10^-2)
= log(6.24) - 2
≈ 0.7952 - 2
= -1.2048
So, the correct expression should be: log(0.0624) = -2 + 0.7952 = -1.2048
4. Role of Characteristic and Mantissa:
- The characteristic (-2) tells us that this number is between 10^-2 and 10^-1.
- The mantissa (0.7952) gives us the precise value.
5. Why It Seems "Far Off from Its True Value":
Your confusion might stem from a misunderstanding of the characteristic. The characteristic isn't simply the integer part of the logarithm; it represents the order of magnitude of the number. In this example, 0.0624 is indeed between 10^-2 and 10^-1, so the characteristic is -2.
6. Estimating Common Logarithms:
Yes, the concepts of characteristic and mantissa can indeed be used to estimate common logarithms. By quickly determining the characteristic, we can get a rough estimate, and then use the mantissa for precise calculation.
7. Modern Applications:
In modern computation, we typically don't need to manually separate the characteristic and mantissa, as calculators and computers can provide exact results directly. However, understanding these concepts is helpful for a deeper comprehension of logarithmic properties and historical calculation methods.
In summary: The concepts of characteristic and mantissa originated from early computational needs. While not commonly used in modern calculations, they provide a method for understanding and estimating logarithms. The expression in your example might have led to some confusion, but in reality, the characteristic and mantissa together form the accurate logarithmic value.
The point of looking at characteristics and mantissas is not that they're "far off from the true value," but rather that they provide a structured way to represent and work with logarithms, especially in the context of historical computation methods and logarithm tables.
In your example, log(0.0624) = -1.2048, we can break this down as:
- Characteristic: -2 (because 0.0624 is between 10^-2 and 10^-1)
- Mantissa: 0.7952 (the positive fractional part)
The full logarithm is then calculated as: -2 + 0.7952 = -1.2048
This method was particularly useful when working with logarithm tables, as it allowed for easier lookup and interpolation of values.
I hope this explanation helps clarify the concepts of characteristic and mantissa, and their application in logarithmic calculations. If you have any further questions, please don't hesitate to ask!

Thank you so much!!!!! This actually answered all my questions!!!!£

Do note that it looks like a ChatGPT answer, though I didn't find anything wrong upon a 5-second skim.
(Sidenote: Being a human myself, I'm cautiously skeptical with ChatGPT, but it's often a good enough starting point.)
I have never heard of characteristic and mantissa before in my how many years in maths (indicating how useful it actually is). But my guess as to why your textbook introduces these concepts is, again, showing logarithm as a powerful tool to measure numbers on an exponential scale (which, for the most part, 12345 is not that different from 34567).
I would suggest not to delve in it too much, as it's really not useful since, as you've suspected, calculating what log(blah) is is easy and interpreting it directly is even easier. Knowing the definition of log (and later, log rules) is way more important IMO.