f-x-y-f-x-f-y-x-R-f-1-8-f-2-3-

Question Number 118118 by naka3546 last updated on 15/Oct/20

f (x + y) = f (x) f (y) \forall x \in R

f (1) = 8

f (\frac{2}{3}) = ?

Answered by mr W last updated on 15/Oct/20

f (x) = a^{x}

f (1) = a^{1} = 8 \Rightarrow a = 8

f (x) = 8^{x}

f (\frac{2}{3}) = 8^{\frac{2}{3}} = 4

Commented by bemath last updated on 15/Oct/20

s i r, h o w d o y o u c l a i m t h a t f (x) = a^{x} ?

Commented by MJS_new last updated on 15/Oct/20

a^{x + y} = a^{x} a^{y}

Commented by bemath last updated on 15/Oct/20

o k a y p r o f t h a n k y o u

Answered by mindispower last updated on 15/Oct/20

f (\frac{2}{3}) = f {(\frac{1}{3})}^{2}

f (\frac{2}{3} + \frac{1}{3}) = f (1) = f (\frac{2}{3}) . f (\frac{1}{3}) = f {(\frac{1}{3})}^{3} = 8 \Rightarrow f (\frac{1}{3}) = 2

f (\frac{2}{3}) = 2^{2} = 4

Answered by aleks041103 last updated on 15/Oct/20

f (x + y) = f (x) f (y)

l n (f (x + y)) = l n (f (x) f (y))

l n (f (x + y)) = l n (f (x)) + l n (f (y))

g (x) = l n (f (x))

\Rightarrow g (x + y) = g (x) + g (y)

l e t y = 0 \Rightarrow g (x) = g (x) + g (0) \Rightarrow g (0) = 0

g (\underset{i = 1}{\sum^{n}} x_{i}) = g (x_{n} + \underset{i = 1}{\sum^{n - 1}} x_{i}) = g (x_{n}) + g (\underset{i = 1}{\sum^{n - 1}} x_{i})

b y i n d u c t i o n

g (\underset{i = 1}{\sum^{n}} x_{i}) = \underset{i = 1}{\sum^{n}} g (x_{i})

i f x_{i} = x

g (n x) = n g (x), n i s i n t e g e r

l n (f (n x)) = n l n (f (x)) = l n ({(f (x))}^{n})

\Rightarrow f (n x) = {(f (x))}^{n}

\Rightarrow f (3 \times \frac{1}{3}) = {(f (\frac{1}{3}))}^{3} = 8

\Rightarrow f (1 / 3) = 8^{1 / 3}

f (2 \times \frac{1}{3}) = {(f (\frac{1}{3}))}^{2} = 8^{2 / 3}

\Rightarrow f (\frac{2}{3}) = 4 1^{2 / 3} = 4 {(e^{2 k π i})}^{2 / 3}

\Rightarrow f (\frac{2}{3}) = 4 e x p (\frac{4 k}{3} π i) = 4 c o s (240 ° k) + 4 s i n (240 ° k) i

\Rightarrow f (2 / 3) = 4; - 2 + 2 \sqrt{3} i; - 2 - 2 \sqrt{3} i