   SEARCH HOME Math Central Quandaries & Queries  Question from keith, a student: please help me solve these two problems for the retake on a test that i have on Monday find the domain: F(x)= 1/ {ln (x+5) - ln (7-x)} find the inverse: g(t)= sqrt {2-3 ln (1-t)} these problems are from the original test (i failed) that i am having the most difficulty with if someone could please show me how to do these problems and explain it by the retake on Monday (4/14/08) i would be so thankful Hi Keith,

Sorry to hear you did poorly on the test.

For your first question you know that the domain of the log function is all positive real numbers and hence you must have x + 5 > 0 and 7 - x > 0. Hence x > -5 and x < 7, or said said in one statement -5 < x < 7. But there is one other possible problem. If ln (x+5) = ln (7-x) then the denominator of F(x) is zero which is not allowed. But (x+5) = ln (7-x) when x + 5 = 7 - x or 2x = 2 which is x = 1. Thus the domain of F(x) is all x which satisfy -5 < x < 7 except x = 1.

For the second problem I am going to write s = sqrt{2 - 3 ln(1 - t)} so then the task is to solve for t.

s2 = 2 - 3 ln(1 - t)
3 ln(1 - t) = 2 - s2
ln(1 - t) = (2 - s2)/3

Raise each side to the power e to get

eln(1 - t) = e(2 - s2)/3
1 - t = e(2 - s2)/3
t = 1 - e(2 - s2)/3

Hence the inverse of g(t)= sqrt {2-3 ln (1-t)} is h(s ) = 1 - e(2 - s2)/3.

I hope this helps and good luck on the retake,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.