Solve 3 · 2^x = 3072
Exponential equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = 10 |
Step-by-step solution
5 steps-
1 Given3 \cdot 2^{x} = 3072
Solve for x.
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2 Isolate the exponential term2^{x} = 1024
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3 Write the right side as a power of 2
Powers with different bases are hard to compare. Facts like 8 = 2³ and 1/4 = 2⁻² rewrite every power with the same base 2; then the exponents can be compared, or the power replaced by a single unknown.
2^{x} = 2^{10} -
4 Same base on both sides, so the exponents must be equal
An exponential function with a positive base other than 1 never takes the same value twice, so bᵐ = bⁿ can only be true when m = n.
x = 10 -
5 Check\begin{aligned}x = 10:\quad 3072 = 3072\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 10 | 3072 | 3072 | ✓ |
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