I will demonstrate two different ways to solve the equation 5^(2x+1) = 25.
Method 1: Taking the logarithm of both sides
Step 1: Start with the equation: 5^(2x+1) = 25
Step 2: Take the logarithm of both sides. We can use any logarithm base, but let's use the natural logarithm (ln) for this example:
ln(5^(2x+1)) = ln(25)
Step 3: Apply the logarithmic property to bring down the exponent:
(2x+1) * ln(5) = ln(25)
Step 4: Divide both sides by ln(5) to solve for x:
2x+1 = ln(25) / ln(5)
Step 5: Subtract 1 from both sides:
2x = (ln(25) / ln(5)) - 1
Step 6: Divide both sides by 2 to isolate x:
x = ((ln(25) / ln(5)) - 1) / 2
Now you can evaluate the right side of the equation to find the numerical value of x.
Method 2: Using the properties of exponents
Step 1: Start with the equation: 5^(2x+1) = 25
Step 2: Rewrite 25 as a power of 5:
5^2 = 25
Step 3: Set the exponents equal to each other:
2x+1 = 2
Step 4: Subtract 1 from both sides:
2x = 2 - 1
Step 5: Divide both sides by 2 to solve for x:
x = (2 - 1) / 2
Simplifying further:
x = 1/2
So, the solutions for the equation 5^(2x+1) = 25 are x = ((ln(25) / ln(5)) - 1) / 2 and x = 1/2.