For IIT aspirants #1

Algebra Level 3

the number of roots of the equation $2^x+2^{x-1}+2^{x-2}= 7^x+7^{x-1}+7^{x-2}$ is

The answer is 1.

2 solutions

Akshat Sharda
Nov 6, 2015

$2^x+2^{x-1}+2^{x-2}= 7^x+7^{x-1}+7^{x-2} \\ 2^{x}\left(\frac{7}{4}\right)=7^{x}\left(\frac{57}{49}\right)\Rightarrow \left(\frac{2}{7}\right)^{x}=\frac{288}{343}\Rightarrow x=\log_{\frac{2}{7}}\left(\frac{228}{343}\right) \\ \text{Only }\boxed{1} \text{ solution.}$

nice solution!!!

Atul Shivam - 5 years, 7 months ago

Thank you !!

Akshat Sharda - 5 years, 7 months ago

besides this can this be solved by another way???

Atul Shivam - 5 years, 7 months ago

@Atul Shivam – I think that this is the most appropriate way.

Akshat Sharda - 5 years, 7 months ago

@Akshat Sharda – I am in search of different solutions besides this which you have posted

Atul Shivam - 5 years, 7 months ago

@Akshat Sharda – that is also good but I want something different :-)

Atul Shivam - 5 years, 7 months ago

Shiban Atif Khan
Nov 23, 2015

Take 2^x-2 common from LHS and 7^x-2 from RHS. Now put the same powered variable aside, take that x-2 power common and then solve using log.

For IIT aspirants #1

The answer is 1.

2 solutions

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