logarithmic and exponential questions

2008-06-05 8:50 pm

1.5logX + 3logX^2

2. 1/3(log[base2]X -log[base2]Y)

3. log7(3X-2)^2

4. log[base 4] (3XYZ)^2

5. log[base5]Y- 4(log[base5]R + 2 log[base5]T)

6.solve for X: log[base b]36 -log[base b]2 = log[base b] X

please explain in details and show all the work!! thank you very much !!

更新1:

7.solve for X correct to four decimal places : e^(2x) = 9.7 8. slove for X: 27^(2x+1) = 9^(4x) 9. write a real life problem that could be solved using the formula for interest compounded continuously, A = Pe^(rt) , and solve it

更新2:

10. the equation for radioactive decay is p=(0.5)^(t/H), where p is the part of a substance with half-life H remaining radioactive after a period of time, t. A given substance has a half-life of 6000 years. after t years, one-fifth of the original sample remains radioactive. find t .

更新3:

11. slove for x: log[base 4] (x^3 +3x)- log[base 4] (x+5) =1 12. solve for m: 3^(m+1) - 5 = 22 13.solve the system of equations: x^2 +y^2 =25 ; 3y - 4x =0

回答 (1)

用戶10012

2008-06-06 12:43 am

✔ 最佳答案

Q.6 Assumed base b.
log36 - log2 = logX . Since logA - logB = log(A/B), that is
log(36/2) = log X
18 = X.
Q.7 e^(2x) = 9.7 Since lnA^B = BlnA, therefore, ln e^(2x) = 2xln e = 2x, that is
2x = ln9.7, therefore, x = (ln9.7)/2.
Q.8
27^(2x + 1) = 9^(4x)
[3^3]^(2x + 1) = [3^2]^(4x)
3^(6x + 3) = 3^(8x)
6x + 3 = 8x
x = 3/2.
Q.11 Assumed base 4.
log(x^3 + 3x) - log(x + 5) = 1 = log 4
(x^3 + 3x)/(x + 5) = 4
x^3 + 3x = 4( x +5) = 4x + 20
x^3 -x -20 = 0
x = solutions to this cubic equation.

2008-06-05 16:49:58 補充：
Q.12
3^(m + 1) - 5 = 22
3^(m + 1 ) = 22 + 5 = 27 = 3^3
m + 1 = 3
m = 2.

2008-06-07 08:11:37 補充：
Q.10 p=0.5^(t/H), p/5 = 0.5^(T/H), second one divided by the first one, we get 0.2=0.5^(T/H -t/H).
Therefore, log0.2 = (T/H - t/H)log0.5, (T-t)/H = log0.2/log0.5, T-t = Hlog0.2/log0.5 =6000log0.2/log0.5=13932. That is after 13932 years, p changes to p/5.

2008-06-07 08:18:03 補充：
Q.13 3y-4x=0 that is y=4x/3, substitute this into the other equation we get x^2 +(4x/3)^2 = 25
x^2 + 16x^2/9 = 25, 9x^2 + 16x^2 = (25)(9), 25x^2 = (25)(9),x^2 = 9, therefore, x =+/-3.And
y= +/-4.

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