math log問題!!(15分)好急!!

1) Explain why 1 + log(x) = log(10x) for every positive number x.

L.H.S.
= 1 + log(x)
= log(10) + log(x)
= log(10x)
= R.H.S.


= = = = =
2) Explain why 2 - log(x) = log (100/x) for every positive number x.

L.H.S.
= 2 - log(x)
= 2log(10) - log(x)
= log(10)² - log(x)
= log(100) - log(x)
= log(100/x)
= R.H.S.

= = = = =
3) Pretend that you are living in the time before calculators and computers existed, and that you have a book showing the log of 1.001, 1.002, 1.003 and so on, up to the log of 9.999. Explain how would you find the log of 457.2, which is beyond the range of you book.

log(457.2)
= log(10x10x4.572)
= log(10) + log(10) + log(4.572)
= 2 + log(4.572)

(log(4.572) can be found in the book.)


= = = =
4)Show that if f is a function with exponential growth, then so is the square root of f. More precisely, show that if f is a function with exponential growth, then so is the function g defined by g(x) = √[f(x)].
f is a function with exponential growth:
f(x) = A•a^(bx)
where A, a and b are constants.

g(x)
= √[f(x)]
= √[A•a^(bx)]
= (√A)•a^(bx/2)

Hence, g(x) is also a function with exponential growth.

2011-03-02 10:37:06 補充：
To: Tak Kin

In Q.3, 3 x 5 x 5 x 6.1 = 457.5 ≠ 457.2

1)log A + log B = log (AxB)...唔洗寫 1 + log(x) = log (10x)Left side = 1 + log(x) = log 10 +log x = log (10x) = Right side 2) log A - log B = log (A/B) , log 100 =2 , )...唔洗寫 Left side = 2 - log(x) = log 100 - log(x) = log (100/x) = Right side 3) log 457.2 = log (3x5x5x6.1) = log 3 + log 5 + log 5 + log 6.1 = log3 + 2log 5 + log 6.1 or log 457.2 = log (3x5x5x6.1) = log3 + log (5^2) + log 6.1 = log3 + 2log 5 + log 6.1 for log A^n = nlog A 上面是解釋,功課咁答足夠log 457.2 = log (3x5x5x6.1) = log3 + 2log 5 + log 6.1 又log3 , log 5 , log 6.1 數值已知(Given)第四題唔識 又log 10 = 1 , log 100 = log 10^2 = 2log10= 2, log 1000 = log 10^3 = 3 log10 =3log10^n = n log10 = n