Introduction / Context:
This is an exponential equation involving powers of 2. The exponents differ by a constant, which suggests factoring out a common power of 2. Solving such equations efficiently is a key skill in algebra and aptitude tests.
Given Data / Assumptions:
- Equation: 2^(x + 4) − 2^(x + 2) = 3.
- Base of the exponents is 2, a positive number.
- We need to solve for the real value of x.
Concept / Approach:
Important ideas:
- Factor out the smaller power of 2 from both terms to simplify the equation.
- Use exponent rules: 2^(x + 4) = 2^(x + 2) · 2².
- Transform the equation into a simple linear equation in terms of 2^(x + 2).
- Recognise that 2^k = 1 only when k = 0, which helps solve for x.
Step-by-Step Solution:
Step 1: Rewrite 2^(x + 4) as 2^(x + 2) × 2² = 4 · 2^(x + 2).
Step 2: Substitute into the equation: 4 · 2^(x + 2) − 2^(x + 2) = 3.
Step 3: Factor out 2^(x + 2): 2^(x + 2)(4 − 1) = 3.
Step 4: Simplify the bracket: 4 − 1 = 3, so 2^(x + 2) × 3 = 3.
Step 5: Divide both sides by 3: 2^(x + 2) = 1.
Step 6: The equation 2^k = 1 holds only when k = 0 because base 2 is not 1.
Step 7: Therefore x + 2 = 0 and x = −2.
Verification / Alternative check:
Step 1: Substitute x = −2 into the original equation.
Step 2: Compute 2^(x + 4) = 2^2 = 4 and 2^(x + 2) = 2^0 = 1.
Step 3: Evaluate 4 − 1 = 3, which matches the right side of the equation, confirming x = −2 is correct.
Why Other Options Are Wrong:
Option 0 would give 2^4 − 2^2 = 16 − 4 = 12, not 3.
Option 2 yields 2^6 − 2^4 = 64 − 16 = 48, which is far from 3.
Option −1 gives 2^3 − 2^1 = 8 − 2 = 6, not 3.
Option 1 gives 2^5 − 2^3 = 32 − 8 = 24, again incorrect.
Common Pitfalls:
A common mistake is to treat 2^(x + 4) − 2^(x + 2) as 2^(2x + 6) or combine exponents incorrectly.
Some learners try to take logarithms prematurely rather than factoring, which makes the problem unnecessarily complicated.
Another error is to forget that 2^k = 1 only for k = 0, not for other values.
Final Answer:
The solution of the equation is
x = −2.
Discussion & Comments