Question

If f(x)= 7^x then prove that f(m+n+p) = f(m).f(n).f(p)

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\sf{f(x) = {7}^{x} }$

TO PROVE

$\sf{f(m + n + p ) = f(m).f(n).f(p)\: \: }$

PROPERTY TO BE IMPLEMENTED

We are aware of the property of indices that

$\sf{ {a}^{m} \times {a}^{n} = {a}^{m + n} \: }$

PROOF

Here

$\sf{f(x) = {7}^{x} }$

So

$\sf{f(m + n + p) = {7}^{(m + n + p)} }$

$\sf{f(m) = {7}^{m} }$

$\sf{f(n) = {7}^{n} }$

$\sf{f(p) = {7}^{p} }$

LHS

$= \sf{f(m + n + p) }$

$\sf{ = {7}^{(m + n + p)} }$

RHS

$\sf{= f(m).f(n).f(p)\: \: }$

$\sf{= {7}^{m} \: . \: {7}^{n} \: . \: {7}^{p}\: }$

$\sf{ = {7}^{(m + n + p)} } \: \: (using \: above \: property \: )$

Hence LHS = RHS

Hence proved

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