Problem 22 - College Algebra

Question

Evaluate each expression without using a calculator. $$ \log _{7} 49 $$

Step-by-Step Solution

Answer

The evaluation for the given logarithmic expression \(\log _{7} 49\) is 2.

Step 1: Understand the Problem

The expression \(\log _{7} 49\) is asking for the exponent that, when used on the base 7, results in 49. In other words, if you have 7 to the power of x equals 49, what is the value of x?

Step 2: Apply Logarithmic Rule

We know that \(7^2 = 49\). This indicates that 7 has to be raised to the power of 2 to result in 49. Therefore, in \(\log _{7} 49\), the result is 2.

Key Concepts

Understanding Logarithms

To solve problems involving logarithmic expressions, it's essential to understand what a logarithm represents. A logarithm answers the question, 'To what exponent must the base be raised, to produce a certain number?' For instance, when you encounter an expression like \(\log_{b}a\), you're searching for the exponent \(x\) that makes the equation \(b^x = a\) true. It's like a detective game where \(b\) and \(a\) are clues, and you're trying to find the missing number \(x\) that links them together in an exponential relationship.

In our exercise, \(\log_{7}49\), we want to find the power to which the number 7 must be raised to get 49. This is a direct application of the definition of a logarithm. The relationship between logarithms and exponents is fundamental in understanding how to deal with such algebraic expressions.

The Power of Exponents

Exponents are like shortcuts in mathematics, compactly expressing repeated multiplication. For example, instead of writing \(2 \times 2 \times 2 \times 2\), you can express it as \(2^4\). Here, 2 is the base, and 4 is the exponent, which tells us how many times the base is multiplied by itself. This compact form is incredibly powerful, especially when dealing with large numbers or complex mathematical problems.

Understanding exponents is crucial for solving logarithmic expressions, as logarithms and exponents are interconnected. The expression \(7^2 = 49\) shows that 7 raised to the 2nd power equals 49, which directly leads us to the solution to our problem \(\log_{7}49 = 2\). This simple equation beautifully demonstrates how logarithms and exponents are two sides of the same coin.

Algebraic Problem-Solving Strategies

When facing algebraic problems, especially those with logarithms and exponents, it's good to have a clear strategy. Here's an approach you can take:

• Firstly, clarify the question to make sure you understand what's being asked.
• Then, identify the relevant rules or patterns that apply to the problem. For logarithms, this means understanding the log rules and how they relate to exponents.
• Afterward, apply these rules step by step to simplify the expression or solve for the variable.
• Finally, check your work by substituting your answer back into the original equation to see if it makes sense.

By breaking down the problem into smaller, manageable parts and systematically applying rules, the process becomes more approachable. With practice, you'll be able to tackle increasingly complex algebraic expressions with confidence.