Need help on 3 math problems?

Question

Hi, i don't know how to do these 3 problems,please give me a help
1. Solve for x:
log 5 (5^2x) = 8

2. log x^3/2 - log (square root x) = 5

3. ln (1/x) + ln (2x^3) = ln 3

For number 1 the answer on the back of the book is 4,#2 is 10^5,and #3 is square root 3/2,i can't get any of those answers,please tell me how to get those answers.
Showing all the steps will be better for me to understand completely,thanks!!

Answer 1

#1: Okay, first of all, when indicating a subscript on computer you generally use an underscore, like this: log_5. Also, when both the 2 and the x are supposed to be part of the exponent, you should put parentheses around them seperately - the reason is that exponentiation has a higher priority than multiplication, and so 5^2x means 25x, not 5^(2x). As to the problem itself, remember that:

log (x^y)=y log x
log_x x = 1 (since x^1 = x).

Thus:

log_5 (5^(2x))=8
2x log_5 5 = 8
2x=8
x=3.9999.... (repeating)

Sorry, long day - although you should give that as an answer to your teacher and see whether she marks it wrong... that's gotta be worth a few laughs at least (said by a verteran of the .99999... = 1 wars).

#2: I'd admonish you about the parentheses thing again, only in this case, you got it right by chance, since (log (x³))/2=log (x^(3/2)). Anyway, you have 3/2 log x - 1/2 log x = 5 (remember, √x = x^(1/2)), thus log x = 5. Exponentiate both sides to get x = 10^5 = 100,000

#3: ln (xy)=ln x + ln y, thus:

-ln x + ln 2 + 3 ln x = ln 3 (remember, 1/x=x^(-1))
2 ln x = ln 3 - ln 2
using the laws in reverse:
ln (x²) = ln (3/2)
exponentiating:
x²=3/2
x=√(3/2) (must be positive, since the logarithim of a negative number doesn't exist in the reals).

Answer 2

1. log 5(5^2x) = 8
log5^(2x+1) = 8 then: (2x+1)log5 = 8
2x+1 = 8/log5 then : 2x =8/log5 - 1
x = (8/log5 -1 )/2

2.log x^3/2 - log x^1/2 =5
3/2 log x - 1/2 log x = 5
log x =5 then X= 10^5


3.ln (1/x) + ln (2x^3) = ln3
ln x^(-1) + ln 2 + ln x^3 = ln3
-lnx + 3lnx = ln3 - ln2
2 ln x = ln (3/2)
ln x^2 = ln (3/2) so: x^2 = 3/2 then
x= square root (3/2)

Answer 3

1. Log 5 (5^2x) is simply 2x. This gives 2x = 8, so x = 4.

2. log x^3/2 is ambiguous. Assuming log x^(3/2), and exponentiating both sides, we have x^(3/2) / x^(1/2) = 10^5. Simplifying the left side gives us x = 10^5.

3. Exponentiate both sides, to get:
(1/x) (2x^3) = 3 Simplify the left side, to get:
2x^2 = 3. Then x = sqrt(3/2). Voila!

Answer 4

[1]
definition: if log(base b) of c = a, then b^a = c

log(base 5) of (5^(2x)) = 8
5^8 = 5^(2x)
8 = 2x
x = 4

[2]
definition: log (a) - log (b) = log (a/b)
definition: sqrt(a) = a^(1/2)
definition: (a^x) / (a^y) = a^(x-y)

log (x^(3/2)) - log(x^(1/2)) = 5
log [(x^(3/2))/(x^(1/2))] = 5
log (x^(3/2 - 1/2)) = 5
log (x^1) = 5
because no base is given for the logarithm, assume base-10 common log
log(base 10) x = 5
10^5 = x

[3]
definition: log(a) + log(b) = log(a * b)

ln(1/x) + ln(2x^3) = ln(3)
ln((1/x) * (2x^3)) = ln(3)
ln(2x^2) = ln(3)
2x^2 = 3
x^2 = 3/2
x = sqrt(3/2)

Answer 5

1.5^8=5^2x
8=2x
x=4