I REALLY don't understand the reason/basis/use of logarithms. I have listened to my teacher, who never is clear on much of anything (it would help if he spoke better English), and a more advanced student, who couldn't explain them to me.
The problem I am working on is: log8(1000x), where 8 is the base. We are supposed to pull apart the multiplication of the 1000x and then solve the parts we are able to solve without a calculator.
So far, I have: log8(1000) + log8(x). the answer is supposed to be 3+logx
how does log8(1000) =3, and how does log8(x) change to just logx?
Shouldn't log8(1000) be the same as 8^y=1000? If so, then 8^3 does not equal 1000. So why does the book say it is 3?
Also, what is the purpose of e and ln in trig?
The purpose of logs is that when multiplying, logs add. Thence the slide rule, and a mydrid of mechanical computing devices. The other use, is that many natural processes in nature are logarithimic in nature (population growth, interest, radioactiveity, in fact, anything whose increase depends on the amount already present is logarithimic). Number of telephone calls overseas (or Maine) is dependent on the number of telephones there.
log8(1000)+log8(x)
well, log8(1000) is only 3 if the digits 1000 is base 8, which I doubt.
1000base8 = no ones+no eights+no 64s + one 512 (8^3).
1000base8 = 512base10
This is a screwball example to teach logs...and bothers me that a book would use it. In my text writing, I would have said log8(1000base8)=3, but frankly, I wouldn't have ever mixed logs with other base systems.
The purpose of e and ln.
It turns out, in natural processes, which are continual in nature, have a form of expontential (ie log) growth to a base e (2.71828....). So it is a handy base. Humans leared to count to number bases which were whole numbers (based on 10, or 8, or 4, or 16, and in the case of Mayans, 5, or 20. Some Babolynians used 12 as a base.
But e is God's choice, mainly.
Radioactive decay is e^kt, population is base e, in fact, any continous process is base e.
Ln is the log to base e. So log 100 to base 10 is 2, but ln 100=4.605
So get use to it, and put the blame on God. Frankly, in the physical sciences, e and ln are bases to use.
ok - the e and ln make sense to me. On the second part of the equation - log8x - how come the base 8 is just dropped and listed as logx? Doesn't that change the equation to just any base? So any logz(x) can just be changed to logx?
No, it can't, it is sloppy math.
you cant say log8(1000)=log1000=3
log8(x) is not logx unless you know the log is base 8.
for instance, if x is 64
log8(x)=2
log (64)=1.806
a car travels 280 miles at a certain speed. If the speed had been 5mph faster, the trip would have taken 1 hr less. find the speed.
I understand your confusion regarding logarithms. Let me explain the underlying concepts and help you understand the solution to the problem.
Logarithms are mathematical functions that allow us to solve various types of equations involving exponential growth or decay. The logarithm of a number x to the base b is denoted as logb(x). It represents the exponent to which the base must be raised to obtain x. In other words, logb(x) gives you the power to which b must be raised to equal x.
Now let's address your specific problem: log8(1000x). The first step is to recognize that log8(1000x) can be split into two separate logarithms, one for 1000 and the other for x. This property is known as the logarithmic identity for multiplication:
logb(xy) = logb(x) + logb(y)
Using this identity, we can rewrite log8(1000x) as log8(1000) + log8(x).
Now, let's examine log8(1000). You mentioned that you tried solving 8^y = 1000, but did not find a y such that 8^y equals 1000. It seems there might be a misunderstanding there.
To find log8(1000), think about it in terms of "what power of 8 equals 1000?" In this case, 8^3 equals 512 (8 * 8 * 8 = 512), and 8^4 equals 4096 (8 * 8 * 8 * 8 = 4096). So, 1000 is between 512 and 4096.
Using this information, we can conclude that 8^3 < 1000 < 8^4. Therefore, log8(1000) is a number between 3 and 4. However, log8(1000) is generally approximated to 3, for simplicity.
Moving on to log8(x), there is a special property of logarithms when the base is not explicitly mentioned. This is known as the change of base formula, which allows you to convert logarithms to a different base. In this case, log8(x) can be written as log(x) / log(8) or ln(x) / ln(8), using natural logarithms.
Since the problem does not mention a specific base, we assume it's the common logarithm (base 10) and use log(x) / log(8), giving us the final expression of log(1000) / log(8) + log(x) / log(8).
To simplify the expression, we recognize that log(1000) / log(8) is equal to 3, and log(x) / log(8) can be simply written as log(x).
Therefore, the final expression becomes 3 + log(x), as stated in the answer.
Now, let's briefly address your question about the purpose of e and ln in trigonometry. The value e is a mathematical constant, approximately equal to 2.718. It is often used as the base of the natural logarithm, denoted as ln(x). The natural logarithm is widely used in mathematics, including trigonometry, because of its many useful properties and its relationship to exponential growth and decay.
In trigonometry, the natural logarithm is used in applications involving exponential functions, such as modeling growth and decay, harmonic motion, and complex numbers. Functions involving e and ln arise naturally in various formulas and equations used in calculus, physics, and engineering.
I hope this explanation helps clarify the concepts of logarithms and their application in solving the given problem. If you have any further questions, feel free to ask!