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

$$f(x+y)=f\left(x\right)f\left(y\right)\mathrm{\forall }x\in \mathbb{R}$$$$f\left(1\right)=8$$$$f\left(\frac{2}{3}\right)=?$$

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

$$f\left(x\right)=a^x$$$$f\left(1\right)=a^1=8\Rightarrow a=8$$$$f\left(x\right)=8^x$$$$f\left(\frac{2}{3}\right)=8^{\frac{2}{3}}=4$$

Commented by bemath last updated on 15/Oct/20

$$sir,howdoyouclaimthatf\left(x\right)=a^x?$$

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

$$okayprofthankyou$$

Answered by mindispower last updated on 15/Oct/20

$$f\left(\frac{2}{3}\right)=f{\left(\frac{1}{3}\right)}^2$$$$f(\frac{2}{3}+\frac{1}{3})=f\left(1\right)=f\left(\frac{2}{3}\right).f\left(\frac{1}{3}\right)=f{\left(\frac{1}{3}\right)}^3=8\Rightarrow f\left(\frac{1}{3}\right)=2$$$$f\left(\frac{2}{3}\right)=2^2=4$$$$$$

Answered by aleks041103 last updated on 15/Oct/20

$$f(x+y)=f\left(x\right)f\left(y\right)$$$$ln\left(f(x+y)\right)=ln\left(f\left(x\right)f\left(y\right)\right)$$$$ln\left(f(x+y)\right)=ln\left(f\left(x\right)\right)+ln\left(f\left(y\right)\right)$$$$g\left(x\right)=ln\left(f\left(x\right)\right)$$$$\Rightarrow g(x+y)=g\left(x\right)+g\left(y\right)$$$$lety=0\Rightarrow g\left(x\right)=g\left(x\right)+g\left(0\right)\Rightarrow g\left(0\right)=0$$$$g\left(\underset{i=1}{\sum ^n}{x}_i\right)=g(x_n+\underset{i=1}{\sum ^{n-1}}{x}_i)=g\left(x_n\right)+g\left(\underset{i=1}{\sum ^{n-1}}{x}_i\right)$$$$byinduction$$$$g\left(\underset{i=1}{\sum ^n}{x}_i\right)=\underset{i=1}{\sum ^n}g\left(x_i\right)$$$$if{x}_i=x$$$$g\left(nx\right)=ng\left(x\right),nisinteger$$$$ln\left(f\left(nx\right)\right)=nln\left(f\left(x\right)\right)=ln\left({\left(f\left(x\right)\right)}^n\right)$$$$\Rightarrow f\left(nx\right)={\left(f\left(x\right)\right)}^n$$$$\Rightarrow f\left(3\times \frac{1}{3}\right)={\left(f\left(\frac{1}{3}\right)\right)}^3=8$$$$\Rightarrow f\left(1/3\right)=8^{1/3}$$$$f\left(2\times \frac{1}{3}\right)={\left(f\left(\frac{1}{3}\right)\right)}^2=8^{2/3}$$$$\Rightarrow f\left(\frac{2}{3}\right)=41^{2/3}=4{\left(e^{2k\pi i}\right)}^{2/3}$$$$\Rightarrow f\left(\frac{2}{3}\right)=4exp\left(\frac{4k}{3}\pi i\right)=4cos\left(240°k\right)+4sin\left(240°k\right)i$$$$\Rightarrow f\left(2/3\right)=4;-2+2\sqrt{3}i;-2-2\sqrt{3}i$$

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