Question

solve my question:-
${2}^{x + 1} \times {2}^{x - 1} = 320$
​

Answer 1

Answer:

$2^{x+1}\times \:2^{x-1}=320\quad :\quad x=\dfrac{\ln \left(320\right)}{2\ln \left(2\right)}\quad \left(\mathrm{Decimal}:\quad x=4.16096\dots \right)$

Step-by-step explanation:

$2^{x+1}\times \:2^{x-1}=320$

$\mathrm{Apply\:exponent\:rule}:\quad \:a^b\times \:a^c=a^{b+c}$

$2^{x+1}\times \:2^{x-1}=2^{x+1+x-1}$

$2^{x+1+x-1}=320$

$\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)$

$\ln \left(2^{x+1+x-1}\right)=\ln \left(320\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\times \log _a\left(x\right)$

$\ln \left(2^{x+1+x-1}\right)=\left(x+1+x-1\right)\ln \left(2\right)$

$\left(x+1+x-1\right)\ln \left(2\right)=\ln \left(320\right)$

$\mathrm{Simplify}$

$2\ln \left(2\right)x=\ln \left(320\right)$

$\mathrm{Solve\:}\:2\ln \left(2\right)x=\ln \left(320\right)$

$2\ln \left(2\right)x=\ln \left(320\right)$

$\mathrm{Divide\:both\:sides\:by\:}2\ln \left(2\right)$

$\dfrac{2\ln \left(2\right)x}{2\ln \left(2\right)}=\dfrac{\ln \left(320\right)}{2\ln \left(2\right)}$

$\mathrm{Simplify}$

$x=\dfrac{\ln \left(320\right)}{2\ln \left(2\right)}$

Answer 2

${2}^{x \: + \: 1}$ + ${2}^{x \: - \: 1}$ = 320

____________[GIVEN EQUATION]

=> $\dfrac{2x}{+1\:+\:1}$ + 2x (1 + 1) = 320

{If the value of power is in positive then we will divided that value with the given number and add one with it.

Ex :- In above case $\dfrac{2x}{+1\:+\:1}$

Power is x + 1

i.e. positive.

So, we will divided it with 1 (which is its power) and add one in it.

i.e. 1+1 = 2

Similarly in ${2}^{x \: - \: 1}$ power is in negative. So, we will multiply it with the given number.

Given number = 2

While multiplying we will add one with its power too.}

=> $\dfrac{2x}{2}$ + 2x × 2 = 320

• Take 2x common from L.H.S.

=> 2x $(\dfrac{1}{2} \: + \: 2)$ = 320

=> 2x $(\dfrac{5}{2}$ = 320

=> 2x × 5 = 320 × 2

=> 2x × 5 = 640

=> 2x = $\dfrac{640}{5}$

=> 2x = 128

=> 2x = ${2}^{7}$

_____________________________

=> x = 7

____________[ANSWER]