The Correct Answer and Explanation is:
The correct value of x is 5.
To solve the equation 2^x = 32, we must determine the exponent, x, that makes the statement true. This type of problem is known as an exponential equation, where the variable we are solving for is in the exponent. The most direct method to find the solution is to express both sides of the equation using the same base number.
The left side of the equation already has a base of 2. Our goal, therefore, is to rewrite the number 32 as a power of 2. We can achieve this by systematically multiplying 2 by itself until we reach 32. Let’s look at the powers of 2:
- 2^1 = 2
- 2^2 = 2 * 2 = 4
- 2^3 = 2 * 2 * 2 = 8
- 2^4 = 2 * 2 * 2 * 2 = 16
- 2^5 = 2 * 2 * 2 * 2 * 2 = 32
From this calculation, we can see that 32 is equivalent to 2 raised to the power of 5.
Now, we can substitute this value back into our original equation. The equation 2^x = 32 becomes 2^x = 2^5.
A fundamental rule of algebra states that if two exponential expressions are equal and they share the same base, then their exponents must also be equal. In our transformed equation, both sides have the base 2. This allows us to set the exponents equal to each other.
By equating the exponents, we get:
x = 5
To verify our answer, we can substitute 5 back into the original equation for x. This gives us 2^5, which we have already calculated to be 32. Since 32 = 32, our solution is correct. Thus, the value of x that satisfies the equation is 5.
