why is the logarithm of 651 2.813580989?

Question

or the log of 1234 = 3.09131516
i don't understand this logarithm thing, can someone please explain how it works?
thanks!

Answer 1

log A = B

means
10^B = A

thus log 1000 = 3 ; log 100 = 2

but for other numbers that are not powers of 10, the result becomes decimal...


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Answer 2

A logarithm is an exponent used in mathematical calculations to depict the perceived levels of variable quantities such as visible light energy, electromagnetic field strength, and sound intensity.

Suppose three real numbers a, x, and y are related according to the following equation:

x = ay

Then y is defined as the base-a logarithm of x. This is written as follows:

log a x = y

As an example, consider the expression 100 = 102. This is equivalent to saying that the base-10 logarithm of 100 is 2; that is, log10 100 = 2. Note also that 1000 = 103; thus log10 1000 = 3. (With base-10 logarithms, the subscript 10 is often omitted, so we could write log 100 = 2 and log 1000 = 3). When the base-10 logarithm of a quantity increases by 1, the quantity itself increases by a factor of 10. A 10-to-1 change in the size of a quantity, resulting in a logarithmic increase or decrease of 1, is called an order of magnitude. Thus, 1000 is one order of magnitude larger than 100.

Base-10 logarithms, also called common logarithms, are used in electronics and experimental science. In theoretical science and mathematics, another logarithmic base is encountered: the transcendental number e, which is approximately equal to 2.71828. Base-e logarithms, written loge or ln, are also known as natural logarithms. If x = ey, then

log e x = ln x = y

Answer 3

"logarithm" is another name for "exponent". When talking about logarithms, one needs to specify the "base", that is, the number that is being raised to those exponents.

In your examples, it appears that the base is 10. To think about why the answers you have are reasonable, think about powers of 10:

10^2 = 100, so the logarithm (exponent) of 100 is 2
10^3 = 1000 , so the logarithm of 1000 is 3
10^4 = 10,000 , so the logarithm of 10,000 is 4

Now consider your first example. To ask, "what's the logarithm of 651?" is to ask "10 to the what? = 651"

Compare with the little table I made above: since 651 is between 100 and 1000, the required power of 10 is between 2 and 3. And since 651 is a lot closer to 1000 than it is to 100, you would expect its logarithm to be closer to 3 than to 2. A calculator gives 2.8135.......

Likewise, 1234 is between 1000 (which has a log of 3) and 10000 (which has a log of 4) but is only slightly greater than 1000, so its log is only slightly greater than 3.

Answer 4

first you get a calculator. then push . then enter the number you want to take the log of. then push enter. then look at the screen to find the answer.