Problem 249 - Calculus Volume 1

Question

Write the equation in equivalent exponential form. \(\log _{5} 25=2\)

Step-by-Step Solution

Answer

The equivalent exponential form is \( 5^2 = 25 \).

Step 1: Understanding the Logarithmic Equation

We are given a logarithmic equation: \( \log_{5} 25 = 2 \). This means "5 raised to what power equals 25?". As it's given, that power is 2.

Step 2: Applying the Exponential Form

The exponential form of a logarithmic equation \( \log_{b} a = c \) is \( b^c = a \). In our problem, substitute \( b = 5 \), \( a = 25 \), and \( c = 2 \). This implies \( 5^2 = 25 \).

Step 3: Verifying the Exponential Form

We can verify \( 5^2 = 25 \) by calculating the power: \( 5 \times 5 = 25 \), which confirms the equation is correct.

Key Concepts

Logarithmic Equation

A logarithmic equation expresses a relationship between a base number and its power, indicating how many times one must multiply the base by itself to obtain a certain value. When you see an equation like \( \log_{5} 25 = 2 \), it translates to the base number 5 being raised to the power 2 to result in 25.
Logarithmic operations can be incredibly useful for solving problems involving exponential growth or decay, as they allow you to "undo" an exponentiation by retrieving the exponent itself.
Logarithms and exponents are inverse operations, meaning that they can cancel each other out. This relationship is crucial for solving equations where the unknown variable might be an exponent or inside a logarithmic expression.

• If you have a logarithmic equation \( \log_{b} a = c \), it means \( b^c = a \).
• This inverse relationship can help greatly with solving scientific and geometric problems.

Base Conversion

Base conversion involves switching from one base to another, particularly in logarithmic and exponential contexts. For our example, where we identified the base as 5, it's about understanding how this number, when raised to the power of 2, results in 25.
Base conversion can be useful when the need arises to switch logarithms from one base to another, especially if you're working with non-decimal systems.
To change logarithm bases, we commonly use the change of base formula: \[\log_{b} a = \frac{\log_{k} a}{\log_{k} b}\] This formula allows you to evaluate logarithms on your calculator when it's limited to certain bases. For the example given, the logarithmic equation requires no base conversion. However, understanding how to do this can broaden your toolkit for solving various kinds of logarithmic problems.
Remember that in base conversions, rounding can sometimes occur which might lead to slight inaccuracies, so it’s good practice to try to maintain precision wherever possible.
Whenever handling different bases:

• Identify the base and the exponent in the problem.
• Be aware of how switching bases might affect calculations.

Verifying Solutions

Verifying solutions is an important step to ensure the correctness of any mathematical equation or problem. After rewriting the logarithmic equation \( \log_{5} 25 = 2 \) to its exponential form \( 5^2 = 25 \), it's essential to check this calculation.
Here, you confirm that indeed when you multiply 5 by itself, you arrive at 25:
1. Calculate \( 5 \times 5 \) which equals 25. This confirms that our exponential form is correct. 2. Verify by reverting back: take your result, 25, to see if \( \log_{5} 25 = 2 \) holds true.

• Recalculate step-by-step if necessary to catch mistakes.
• Always reflect back to the original setup to maintain alignment.

While this process may seem straightforward, it is a crucial step especially in exams or real-world problems. It helps in reinforcing one's understanding and ensures there are no calculation errors before presenting your final results. Verifying solutions is about double-checking and confidence reassurance.