Question

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ \log _{5} x=2 $$

Short answer

The solution to the equation is \(x = 25\).

Step-by-step solution

Understanding the Logarithmic Form

The exercise involves the logarithmic equation \(\log_{5} x = 2\). According to the hint provided, if \(\log_{a} b = c\), this implies \(a^{c} = b\). In our case, \(a = 5\), \(b = x\), and \(c = 2\).

Applying the Exponential Form

Using the information from the hint, convert the logarithmic equation \(\log_{5} x = 2\) into its exponential form. This means using the equivalence \(a^{c} = b\), so we have \(5^{2} = x\).

Calculating the Exponentiation

Calculate the value of \(5^{2}\). This means multiplying 5 by itself: \(5 \times 5\), which equals 25. Thus, we find that \(x = 25\).

Verification of the Solution

To verify the solution, substitute \(x = 25\) back into the original logarithmic form: \(\log_{5} 25\). Calculate the value to ensure it equals 2. Knowing \(5^{2} = 25\), \(\log_{5} 25 = 2\), confirms that our solution is correct.

Key concepts

Understanding Logarithms

Logarithms are an essential part of mathematics, especially when dealing with exponential equations. A logarithm answers the question: "To what power must we raise a base number to get another number?" For instance, in the equation \(\log_{5} x = 2\), we are trying to find the value of \(x\) for which 5 raised to the power of 2 gives \(x\). This is because the base in this context is 5, and we need to find \(x\) such that \(5^{c} = x\).
The key property of logarithms is their ability to "reverse" exponential functions, much like how subtraction can reverse addition. When you see \(\log_{a} b = c\), it's saying the exponent \(c\) that will result in \(b\) when \(a\) is raised to that power. Understanding this property is crucial when converting a logarithmic expression into an exponential one. This transformation allows for the simplification and solving of equations that involve unknown variables.

Problem Solving with Exponential Equations

Solving exponential equations often involves converting them from their original logarithmic form. The exercise provided, \(\log_{5} x = 2\), is a clear example of this kind of problem. Here the problem asks us to solve for \(x\).
To tackle this, we use the equivalence property of logarithms: \(\log_{a} b = c\) implies \(a^{c} = b\). This means that if you have a logarithm, you can change it into an exponential equation to solve it easily:

• Identify the base of the logarithm. In our case, it's 5.
• Recognize the exponent. Here it's 2.
• Transform the logarithm into the exponential expression: \(5^{2} = x\).

By following these steps, we've effectively turned the problem into a simpler multiplication: \(5 \times 5 = 25\). Hence, solving this helps us find that \(x = 25\). Ensuring you understand this transformation and the resulting arithmetic makes solving these equations straightforward and intuitive.

Mathematical Reasoning and Verification

After solving a problem in mathematics, it's essential to verify your solution to ensure accuracy. Verification is a part of mathematical reasoning, ensuring that your results are logical and correct.
In the context of the provided problem, we solved \(\log_{5} x = 2\) and found \(x = 25\). To verify this solution, substitute \(x\) back into the original logarithmic equation. Calculate \(\log_{5} 25\) and check if it equals 2:

• Recognize that \(5^{2} = 25\). Thus, calculating \(\log_{5} 25\) should indeed give us the exponent 2.
• The result matches our calculated solution, confirming \(x = 25\) is an accurate solution.

Verification is a key step because it not only checks our work but helps reinforce understanding of logarithmic and exponential relationships. This builds confidence and deepens comprehension, making the process of solving such equations smoother in the future.