Question

Can you show me the work to get these answers also?

A.Solve each equation. Leave exact answers:
1.5^3x-2=14
2.log(x-1)-logx=log(x-3)

B.If f(x)=x+2/x and g(x)=7/x-5,
a. Find the composition (f o g)(x) and simplify completely.
b. Find the domain of (f o g)(x).Write answer in interval notation.
c. Find the composition of (g o f)(x) and simplify completely.
d. Find the domain (g o f)(x).Write your answer using interval notation.

Answer 1

1. 0.5^3x-2 = 14
0.5^3x = 14+2 = 16
Take Log of both sides:
3x*Log 0.5 = Log 16
Divide both sides by Log 0.5:
3x = Log 16/Log0.5 = -4
Divide both sides by 3:
X = -4/3

2. Log(x-1)-Log x = Log(x-3)
Log((x-1)/x) = Log(x-3)

B. f(x) = (x+2)/x; g(x) = 7/(x-5)

a. F(g(x)) = (7/(x-5)+2)/(7/(x-5)) =
(7+2(x-5)/(x-5))/(7/(x-5)) =
((2x-3)/(x-5))/(7/(x-5)) =
((2x-3)/(x-5))*(x-5)/7 = (2x-3)/7.

b. Domain: All real values of X.
-Infinity < X < +Infinity.

c. g(F(x)) = 7/((x+2)/x-5)=
7/(x+2-5x)/x = 7/(x+2-5x)/x =
7/(2-4x)/x = 7x/(-2(2x-1))

d. 2x-1 = 0
2x = 1
X = 1/2 Will result in a denominator of zero and the Eq becomes undefined.

DOMAIN: All real values of X except 1/2.
X <> 1/2.