Answer:
(a) x = 9, (9,4)
(b) x = 1, (1,3)
(c) x = 1 and the graph of f(x) and g(x) intersects at point (1,3)
(d) [tex]x = - \frac{53}{3}[/tex] or x = 9
(e) [tex]x = - \frac{33}{23}[/tex]
Step-by-step explanation:
We are given that [tex]f(x) = \log_{2} {(x + 7)}[/tex] ....... (1),and
[tex]g(x) = \log _{2} {(3x + 5)}[/tex] ........ (2)
Now,
(a) We have to solve f(x) = 4
⇒ [tex]f(x) = \log_{2} {(x + 7)} = 4[/tex]
Converting logarithm to exponent form, we get,
[tex]x + 7 = 2^{4} = 16[/tex]
⇒ x = 9 (Answer)
Now, the point on the graph of f(x) will be (9,4) (Answer)
(b) We have to solve g(x) = 3
⇒ [tex]g(x) = \log_{2} {(3x + 5)} = 3[/tex]
Converting logarithm to exponent form, we get,
[tex]3x + 5 = 2^{3} = 8[/tex]
⇒ x = 1 (Answer)
Now, the point on the graph of g(x) will be (1,3) (Answer)
(c) We have to solve f(x) = g(x)
⇒ [tex]\log_{2} {(x + 7)} = \log _{2} {(3x + 5)}[/tex]
Now comparing both sides we can write
x + 7 = 3x + 5
⇒ 2x = 2
⇒ x = 1 (Answer)
Now, at x = 1, [tex]f(x) = \log_{2} {(1 + 7)} = \log_{2} {2^{3}} = 3[/tex]
So, the graph of f(x) and g(x) intersects at point (1,3) (Answer)
(d) We have to solve (f + g)(x) = 9
⇒ [tex]\log_{2} {(x + 7)} + \log _{2} {(3x + 5)} = 9[/tex]
⇒ [tex]\log_{2} {(x + 7)(3x + 5)} = 9[/tex]
⇒ [tex](x + 7)(3x + 5) = 2^{9} = 512[/tex]
⇒ 3x² + 26x - 477 = 0
⇒ (3x + 53)(3x - 27) = 0
Hence, [tex]x = - \frac{53}{3}[/tex] or x = 9 (Answer)
(e) We have to solve (f - g)(x) = 3
⇒ [tex]\log_{2} {(x + 7)} - \log _{2} {(3x + 5)} = 3[/tex]
⇒ [tex]\log_{2} {\frac{x + 7}{3x + 5} = 3[/tex]
⇒ [tex]\frac{x + 7}{3x + 5} = 2^{3} = 8[/tex]
⇒ x + 7 = 24x + 40
⇒ 23x = - 33
⇒ [tex]x = - \frac{33}{23}[/tex] (Answer)
