Question
Solve the equation. \[ \begin{array}{c}2^{x-3}=16 \\ x=[?]\end{array} \]
Ask by Bob Johnson. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 7 \)
Solution
To solve the equation:
\[
2^{x-3} = 16
\]
follow these steps:
### Step 1: Express 16 as a Power of 2
First, recognize that 16 is a power of 2:
\[
16 = 2^4
\]
### Step 2: Set the Exponents Equal
Since the bases are the same (both are 2), you can set the exponents equal to each other:
\[
x - 3 = 4
\]
### Step 3: Solve for \( x \)
Add 3 to both sides of the equation to isolate \( x \):
\[
x = 4 + 3
\]
\[
x = 7
\]
### Final Answer
\[
x = 7
\]
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Simplify this solution The Deep Dive
To solve the equation \(2^{x-3} = 16\), we first recognize that \(16\) can be expressed as a power of \(2\). Specifically, \(16 = 2^4\). Now, we can rewrite the equation: \[ 2^{x-3} = 2^4 \] Since the bases are the same, we set the exponents equal to each other: \[ x - 3 = 4 \] Next, we solve for \(x\) by adding \(3\) to both sides: \[ x = 4 + 3 \] \[ x = 7 \] So the solution is \(x = 7\).
