Question

Consider the following equation \[{{8}^{x-1}}={{2}^{x+3}}\], then x is ______
(a) 2
(b) 4
(c) 1
(d) 3

Answer 1

Hint: First, we will take the LHS of the expression, then we know that 8 can be written as \[{{2}^{3}}\]. Now comparison of the LHS & RHS gives us the fact that both have equal bases. So, we can equate their power and then solve for x.

Complete step-by-step solution -
We have been given an expression, which we need to solve and get the value of x.
The expression given to us is,
\[{{8}^{x-1}}={{2}^{x+3}}\]
Let us first consider the LHS of the expression.
We know that, \[8=2\times 2\times 2={{2}^{3}}\].
\[\therefore \] We can write, \[{{8}^{x-1}}={{2}^{3\left( x-1 \right)}}\].
Now let us put \[{{2}^{3\left( x-1 \right)}}\] in place of \[{{8}^{\left( x-1 \right)}}\] in the given expression.
\[{{8}^{x-1}}={{2}^{x+3}}\]
\[\Rightarrow {{2}^{3\left( x-1 \right)}}={{2}^{x+3}}\]
By the law of exponents RHS and LHS has the same base, thus use can equate the powers of LHS and RHS. Hence, we get,
3 (x – 1) = x + 3
Now, let us simplify the above expression and get the value of x.
3x – 3 = x + 3
3x – x = 3 + 3
\[\Rightarrow \] 2x = 6
\[\therefore x=\dfrac{6}{2}=3\]
Hence we got the value of x = 3.
\[\therefore \] Option (d) is the correct answer.

Note: You can check if your answer is correct by putting x = 3 in the expression, \[{{8}^{x-1}}={{2}^{x+3}}\].
\[\therefore {{8}^{3-1}}={{2}^{3+3}}\Rightarrow {{8}^{2}}={{2}^{6}}=64\]
Thus we can say that for x = 3, the expressions \[{{8}^{x-1}}\] and \[{{2}^{x+3}}\] are equal.