Question

Solve the following equation:
\[{4^x} - {10.2^{x - 1}} = 24\]

Answer 1

Hint:
The approach that can be followed for solving such kind of question is to try express the terms containing a variable in their powers as in their base powers and then searching for the powers with same base and substituting it as a different variable making it an equation which can be solved and the value of x can be calculated further. Particularly in this question firstly express \[{4^x}\]in the powers of the base 2 that is \[{\left( {{2^2}} \right)^x}\]and later breaking \[{2^{x - 1}}\]in fractional form that is \[\dfrac{{{2^x}}}{2}\]simplifying the term and later on substituting \[{2^x}\] as t and later further solving the equation in terms of t for the values of t and further comparing t and \[{2^x}\] for the desired solution that we want.

Complete step by step solution:
Now given equation is
\[{4^x} - {10.2^{x - 1}} = 24\]
Now expressing \[{4^x}\]in the powers of the base 2 that is \[{\left( {{2^2}} \right)^x}\] we get -
\[{({2^x})^2} - {10.2^{x - 1}} = 24\]
Also breaking down \[{2^{x - 1}}\]in the fractional form that is \[\dfrac{{{2^x}}}{2}\]the above equation becomes-
\[{({2^x})^2} - 10.\dfrac{{{2^x}}}{2} = 24\]
Now simplifying the above equation we get –
\[{({2^x})^2} - {5.2^x} = 24\]
Now substituting \[{2^x}\]as t in above equation we get-
\[{t^2} - 5.t = 24\]
\[{t^2} - 5.t - 24 = 0\]
\[{t^2} - 8t + 3t - 24 = 0\]
\[(t + 3)(t - 8) = 0\]
\[(t - 8) = 0\] and \[(t + 3) = 0\]
So the resulting values of t are \[ = - 3\]and \[ = 8\]
Also we have substituted t as \[{2^x}\]
So \[{2^x}\]\[ = - 3\]here x is not real
And also \[{2^x}\]\[ = 8\]here
\[{2^x} = {2^3}\] as bases are same comparing the powers as the rule says
\[
  {t^y} = {t^b} \\
  y = b \\
 \]
\[x = 3\]
So x is equal to \[3\]

Note:
Some common mistakes that a person generally make is fetching the constant in the simpler terms which makes the solution of such kind of questions more complicated following the general rule that is to try to express the terms containing x in their powers as in their base powers and then searching for the powers with same base and substituting it as a different variable making it an equation which can be solved and the value of x can be calculated further this kind of approach can be applied to wider range of questions like this and is much easier to execute.