Solve $5^{2x} = 625$ for x.
Understand the Problem
The question asks us to solve the exponential equation $5^{2x} = 625$ for the variable $x$. We will first express both sides of the equation using the same base, and then equate the exponents to solve for $x$.
Answer
$x = 2$
Answer for screen readers
$x = 2$
Steps to Solve
- Express 625 as a power of 5
We want to express 625 as a power of 5. We can do this by recognizing that $625 = 5 \times 5 \times 5 \times 5 = 5^4$.
- Rewrite the equation
Replace 625 with $5^4$ in the original equation to get $5^{2x} = 5^4$.
- Equate the exponents
Since the bases are equal (both are 5), we can equate the exponents: $2x = 4$.
- Solve for x
Divide both sides of the equation $2x = 4$ by 2 to solve for $x$: $x = \frac{4}{2} = 2$.
$x = 2$
More Information
Exponential equations are equations in which the variable appears in the exponent. To solve an exponential equation where both sides can be written with the same base, we set the exponents equal to each other and solve for the variable.
Tips
A common mistake is failing to recognize that 625 can be written as $5^4$. Another mistake is incorrectly dividing 4 by 2 when solving for $x$.
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