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Chapter 4: Problem 17

Solve the following equations for \(x\). $$5^{2 x}=5^{2}$$

Short Answer

Expert verified
x = 1

Step by step solution

01

Recognize Equal Bases

Identify that both sides of the equation have the same base, which is 5 in this case.
02

Set Exponents Equal

Since the bases are equal, set the exponents equal to each other: \[ 2x = 2 \]
03

Solve for x

Divide both sides of the equation by 2 to solve for x: \[ x = \frac{2}{2} \] Thus, \[ x = 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equal Bases
When solving exponential equations, one key concept is identifying if the bases on both sides of the equation are the same. This is called having 'equal bases'. In our exercise, the equation is \(5^{2x} = 5^2\).
Here, the base is 5 for both exponential terms.
This is crucial because if the bases are equal, it simplifies solving the equation significantly.
It enables us to set the exponents equal to each other, since for the same base, different exponents would mean different values.
Identifying equal bases early on saves time and effort in solving the equation and reduces the possibility of mistakes.
Exponentiation
Exponentiation is the operation of raising a number to a power. In the equation \(5^{2x} = 5^2\), we deal with exponents. Exponents tell us how many times to multiply the base by itself.
For example, \(5^2\) means \(5\times5 = 25\).
When we have an expression like \(5^{2x}\), it means that 5 is multiplied by itself \(2x\) times.
It is important to understand how exponentiation works when solving exponential equations because it directly influences how we manipulate and simplify the terms.
Remember, the laws of exponents, like \((a^m)^n = a^{mn}\) and \(a^m \times a^n = a^{m+n}\), can help in breaking down and rationalizing such expressions.
Equation Solving
Once we have equal bases, the next step is to solve the equation by setting the exponents equal to each other. Let's look at our example again:
Given \(5^{2x} = 5^2\), we know the bases are equal, so we set the exponents equal: \(2x = 2\).
This transforms our exponential equation into a simple linear one.
Now, we just solve for \(x\).
To isolate \(x\), divide both sides by 2: \(\frac{2x}{2} = \frac{2}{2}\).
Simplifying this gives us:
\(x = 1\).
It's important to follow this structured approach - identifying equal bases, equating the exponents, and solving the simplified equation - to confidently and accurately solve exponential equations.

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