find the truth set of set of a. 3^(2x+3) -3^(x+2) -3^(x+1) +1=0 b. log a base4=log b base 8+1
(a) we have here a function f(x) = 3^(2x+3) -3^(x+2) -3^(x+1) +1=0. What we are being asked to find is, in effect, the solution of f(x) = 0. i.e. find those values of x such that 3^(2x+3) -3^(x+2) -3^(x+1) +1=0. This will make the expression/equation true. Considering that the function is using x as an exponent, the simplest way of solving this is to plot the curve of y = 3^(2x+3) -3^(x+2) -3^(x+1) +1, and see where it crosses the x- axis, When plotted, it is noted that the x-axis is crossed at x = -1, and at x = -2. Thus the truth set is:
Answer: x = {-2.-1}
(b) log_4(a) = log_8(b) + 1
Let log_4(a) = x, then a = 4^x
Let log_8(b) = y, then b = 8^y
We can rewrite the eqn log_4(a) = log_8(b) + 1 as x = y + 1, or
x – y = 1
We are creating a truth set in which we are only interested in integers. This makes the eqn x - y = 1 a Diophantine equation for which the solution is,
x = 2 + t
y = 1 + t
t e Z
We can thus write down,
a = 4^(2 + t)
b = 8^(1 + t)
which gives us our truth set.
Answer: {(a, b): a = 4^(2 + t), b = 8^(1 + t), t e Z}