Solving for the Value of an Unknown Variable

Question

The equation is as follows:

$$ \color{red}{4^{2016} - 4^{2015} - 4^{2014} + 4^{2013} = 90(2^x)} $$

My problem is coming from simplifying the left hand side of the equation. I know that there must be an easier way to simplify that side rather than to solve for the individual values of the exponents. After that I can divide by $90$. And another question is how can you take $\log$ of base $2$ of a number (is that how you correctly say it?) on a calculator?

Any help will be greatly appreciated!

Answer 1

Hint:

$$4^{2016} - 4^{2015} - 4^{2014} + 4^{2013} =4^{2013}(4^3-4^2-4+1)$$

Now $4^n=(2^2)^n=2^{2n}$

Answer 2

this equation has one solution.i.e,$x=4025$

Solution:

given, ${4^{2016} - 4^{2015} - 4^{2014} + 4^{2013} = 90(2^x)}$

$\implies4^{2013}(4^3-4^2-4+1)=90(2^x)$

$\implies4^{2013}(64-16-4+1)=90(2^x)$

$\implies4^{2013}(45)=90(2^x)$

$\implies4^{2013}=2^{x+1}$

$\implies2^{4026}=2^{x+1}$

By Equating powers,we get,

$4026=x+1$

Therefore $x=4025$ is only solution