Expert: Sherman D. - 3/30/2008

Question
So I am stumped on how to solve these Logs:
12. log5(x - 1) + log5(x - 2) - log5(x + 6) = 0
13.a) log5(log3x) = 0 *where 3 is subscript
b) log2(log4x) = 1 *where 4 is subscript
14a) 1/2loga(x + 2) + 1/2loga(x - 1) = 2/3loga27 * a is subscript
14b)logb(x - 1) + logb(x + 2) = logb(8 - 2x)

Answer
12.)
log(5)(x - 1) + log(5)(x - 2) - log(5)(x + 6) = 0

using rules of logarithms, this becomes

log(5)(((x - 1)(x - 2))/(x + 6)) = 0
log(5)((x^2 - 2x - x + 2)/(x + 6)) = 0
log(5)((x^2 - 3x + 2)/(x + 6)) = 0
log((x^2 - 3x + 2)/(x + 6)) / log(5) = 0
log((x^2 - 3x + 2)/(x + 6)) = 0
(x^2 - 3x + 2)/(x + 6) = 1
x^2 - 3x + 2 = x + 6
x^2 - 4x - 4 = 0

x = (-b ± sqrt(b^2 - 4ac))/(2a)

x = (-(-4) ± sqrt((-4)^2 - 4(-4)(1)))/(2(1))
x = (4 ± sqrt(16 + 16))/2
x = (4 ± sqrt(32))/2
x = (4 ± sqrt(16 * 2))/2
x = (4 ± 4sqrt(2))/2
x = 2 ± 2sqrt(2)

since you can't have negative logs

x = 2 + 2sqrt(2)

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13.) by this do you mean

a.)
log(base 5)(log(base 3)x) = 0, if so
log(5)(log(x)/log(3)) = 0
log(log(x)/log(3))/log(5) = 0
log(log(x)/log(3)) = 0
log(x)/log(3) = 1
log(x) = log(3)
x = 3

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b.)
log(2)(log(4)x) = 1
log(2)(log(x)/log(4)) = 1
log(log(x)/log(4))/log(2) = 1
log(log(x)/log(4)) = log(2)
log(x)/log(4) = 2
log(x) = 2log(4)
log(x) = log(4^2)
log(x) = log(16)
x = 16

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14.)
a.)
(1/2)log(a)(x + 2) + (1/2)log(a)(x - 1) = (2/3)log(a)27
(1/2)(log(a)(x + 2) + log(a)(x - 1) = log(a)(27^(2/3))
(1/2)(log(a)((x + 2)(x - 1))) = log(a)(9)
log(a)(x^2 - x + 2x - 2) = 2log(a)(9)
log(a)(x^2 + x - 2) = log(a)(9^2)
x^2 + x - 2 = 81
x^2 + x - 83 = 0

x = (-1 ± sqrt(1^2 - 4(1)(-83)))/(2(1))
x = (-1 ± sqrt(333))/2
x = (-1 ± sqrt(9 * 37))/2
x = (-1 ± 3sqrt(37))/2

since you can't have negative logs

x = (-1 + 3sqrt(37))/2

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b.)
log(b)(x - 1) + log(b)(x + 2) = log(b)(8 - 2x)
log(b)((x - 1)(x + 2)) = log(b)(8 - 2x)
x^2 + 2x - x - 2 = 8 - 2x
x^2 + x - 2 = 8 - 2x
x^2 + 3x - 10 = 0
(x + 5)(x - 2) = 0
x = -5 or 2

ANS : x = 2

to check your answers, you can use www.quickmath.com, but you need to convert your problems, for ex:

log(b)(x - 1) + log(b)(x + 2) = log(b)(8 - 2x) whereas b is the subscript would become

(log(x - 1)/log(b)) + (log(x + 2)/log(b)) = (log(8 - 2x)/log(b))