hw help..!! needed before saturday?

Question

Evaluate:
(a)
(¡) log (base3) 9
(¡¡) log (base10)10
(¡¡¡) log (base16) 8

(b) Write log [(8x^4√ 5)/81] in terms of Log2, Log3, and Log5 to any base.
(c) Write (a) log 30 (b) log 450 in term of log 2, log 3 and log 5 to any base.

(d) Evaluate correct to 3 decimal places.
5e ^ -0.982 /3ln 0.0173

Answer 1

(a) A "logarithm" asks the question, "What exponent is needed to get that number?" Ex: 3^2 = 9, so 3 'needs' an exponent of 2.
(i) 2
(ii) 1
(iii) log(16) of 2 = 1/4, so 3/4

(b) Use your properties of logarithms to split it up.
log(8) + 4log(x) + 1/2log(5) - log(81)
3 log(2) + 4 log(x) + 1/2 log(5) - 4 log(3)

(c) 30 = 2 * 3 * 5, so log(30) = log(2) + log(3) + log(5)
450 = 3^2 * 5^2 * 2, so log(450) = log(2) + 2 log(3) + 2 log(5)

(d) Use a calculator.

Answer 2

(a)
(¡) log (base3) 9 = log (base3) 3^2 = 2
(¡¡) log (base10)10 = log (base10) 10^1 = 1
(¡¡¡) log (base16) 8 = log (base16) 16^3/4 = 3/4

Get the pattern? If you can get log (base B) B^x, it equals x. This is due to the inverse property of logs and exponents with the same base.

(b) log [(8x^4√ 5)/81]
= log 8 + log x^4 + log √5 - log 81
= log 2^3 + log x^4 + log 5^(0.5) - log 3^4
= 3log2 + 4log x + 0.5log5 - 4 log 3
But there's a problem here, I have a log x in there and that doesn't follow the instructions. So what I will do is assume the logs are all base x, which would make the term '4 log x ' just equal to 4 (because log (base x) x = 1)
Therefore, the answer is, all in BASE x:
3 log 2 + 4 + 0.5 log 5 - 4 log 3

(C) (i)
log 30 = log (2*3*5) = log 2 + log 3 + log 5 (this can be in any base)
(ii) log 450 = log (2*3*3*5*5) = log 2 + 2 log 3 + 2 log 5 (also can be written in any base)

(d) I preface this problem by saying that I am not sure if you are missing parentheses that would change the answer. I assumed the natural log is actually part of the exponent of the e, not dividing into the e. Thus.....

Ignoring the coefficient of 5 for a moment,
e^(-0.982 /3)ln 0.0173
= e^ln (0.0173^(-0.982 /3))
= (0.0173^(-0.982 /3)) = .265. Then multiply by 5 to get 1.325.

Answer 3

(a)
(¡) log (base3) 9 = 2 because 3^2 =9
(¡¡) log (base10)10 =1 because 10^1=10
(¡¡¡) log (base16) 8 = 3/4 because 16^(3/4) = 8

(b) Write log [(8x^4√ 5)/81] in terms of Log2, Log3, and Log5 to any base.
=log8x^4 +log 5^.5 - log 81
= log2^3 +log x^4 +log 5^.5 - log 3^4
= 3log2 +4logx +.5log5 -4log3

(c) Write (a) log 30 (b) log 450 in term of log 2, log 3 and log 5 to any base.
Log 30 = log2+log 5 +log3
Log450= log 9+log 25 +log2= 3log2 +2log5 +log3


(d) Evaluate correct to 3 decimal places.
5e ^ -0.982 /3ln 0.0173
5e ^ -0.982 = 1.8728 and 3ln0.0173 = -12.171
5e ^ -0.982 /3ln 0.0173 = -0.154

Answer 4

the log is the power to which the base must be raised to equall that number.
The log of any base is 1. (log base 2 of 2 is 1, log base 3 of 3 is 1).
Log base 3 of 9 is 2 because 3^2 = 9.
log base 10 of 10 is 1.
log base 16 of 8 is going to be less than one because the number is less than the base.
log base 16 of two is 1/4 because 2^4 is 16.
log base A of B^C = C times log base A of B.
log base 16 of 8 is 3 times log base 16 of 2 or 3/4.

log base 2 of ((8x^4sqrt5)/81) =
log base 2 of 8 x^4sqrt5 - log base two of 81
log base 2 (8) + 4sqrt5 log base 2 of x - log base 2 of 3^4.
3+4sqrt5 log base 2 of x -4 log base 2 of 3
log base 2 of 3 is 1.585 APPX
- 3.34 + 4sqrt5 log base 2 of x.

log base 3 of ((8x^4sqrt5)/81)
log base 3 of 8 x^4sqrt5 - log base 3 of 81
1.893 + 4sqrt5 log base 3 of x - 4
-3.107 + 4sqrt5 log base 3 of x

log base any of ((8x^4sqrt5)/81)
log base any of 8+4sqrt5 log base any of x - log base any of 81

5 e^-0.982 / 3 ln 0.0173

5 times 2.71828183^-0.982 / -4.05705
-2.14 that is assuming there were not parentheses the divsion was over all not just in the exponent