Step-by-step explanation:
Method 1: Taking the logarithm of both sides:
Step 1: Start with the equation 2^(2x-1) = 25.
Step 2: Take the logarithm of both sides. We can use any base logarithm; here, we'll use the natural logarithm (ln):
ln(2^(2x-1)) = ln(25).
Step 3: Apply the logarithm property to bring down the exponent:
(2x-1) ln(2) = ln(25).
Step 4: Divide both sides of the equation by ln(2):
2x - 1 = ln(25) / ln(2).
Step 5: Solve for x by isolating the variable:
2x = (ln(25) / ln(2)) + 1.
Step 6: Divide both sides by 2:
x = [(ln(25) / ln(2)) + 1] / 2.
So, one possible solution to the equation 2^(2x-1) = 25 is x = [(ln(25) / ln(2)) + 1] / 2.
Method 2: Solving by manipulating the equation:
Step 1: Start with the equation 2^(2x-1) = 25.
Step 2: Rewrite 25 as 2^2 5:
2^(2x-1) = 2^2 * 5.
Step 3: Apply the power property of equality:
2x - 1 = 2 + log₂(5).
Step 4: Simplify the equation:
2x = 3 + log₂(5).
Step 5: Divide both sides by 2:
x = (3 + log₂(5)) / 2.
So, another possible solution to the equation 2^(2x-1) = 25 is x = (3 + log₂(5)) / 2.
Both methods yield a solution, and you can choose the one that is more convenient or suitable for your particular situation.