Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Find the exact value of the logarithmic expression without using a calculator.\(\log _{7} \frac{49}{343}\)

Step-by-Step Solution

Answer

-1

Step 1: Express the fraction in the logarithmic expression as powers of 7

Firstly, express the both the numerator and denominator as powers of 7. 49 can be expressed as \(7^2\) and 343 can be expressed as \(7^3\)

Step 2: Apply the quotient rule of logarithms

The quotient rule of logarithms states that the logarithm of a quotient is equal to the subtraction of the logarithm of the numerator by the logarithm of the denominator. That is, \(\log _{b} \frac{M}{N} = \log _{b} M - \log _{b} N\). So, \(\log _{7} \frac{7^2}{7^3} = \log _{7} 7^2 - \log _{7} 7^3\)

Step 3: Simplify the logarithm

The rules of logarithms state that \(\log _{b} b^x = x\). So, \(\log _{7} 7^2 = 2\) and \(\log _{7} 7^3 = 3\). Therefore, \(\log _{7} 7^2 - \log _{7} 7^3 = 2 - 3\)

Key Concepts

Quotient Rule

The quotient rule in logarithms is a handy tool when dealing with expressions that involve division inside the logarithmic function. Imagine you have a fraction inside a logarithm, like in our original expression \(\log_{7} \frac{49}{343}\). The rule tells us that we can transform this into a subtraction problem, making it much easier to handle.

Here's how it works: the logarithm of a fraction \(\log_{b} \frac{M}{N}\) becomes \(\log_{b} M - \log_{b} N\). Simply put, you find the logarithm of the numerator and then subtract the logarithm of the denominator. This significantly simplifies the process, breaking it down into smaller, more manageable parts.

In our exercise, we applied the quotient rule to transform \(\log_{7} \frac{7^2}{7^3}\) into \(\log_{7} 7^2 - \log_{7} 7^3\). This decomposition sets the stage for the next steps in solving the expression.

Exponents

Understanding exponents is crucial when working with logarithms, as they often appear in the forms of bases and powers. An exponent tells us how many times a number, called the base, is multiplied by itself.

For instance, in our context, 49 can be expressed as \(7^2\), meaning 7 multiplied by itself once (7*7). Similarly, 343 can be expressed as \(7^3\), indicating 7 multiplied by itself twice (7*7*7). This recognition simplifies the numbers and makes it easier to apply rules of logarithms.

By expressing both parts of the fraction \(\frac{49}{343}\) as powers of the same base (7), the logarithmic expression becomes easier to simplify. We can then use the rule \(\log_{b} b^x = x\), which states that the logarithm of a base raised to a power equals the exponent itself. Applying this simplified even further, as you'll see shortly.

Simplifying Expressions

The final step in our problem is simplifying the expression after applying the rules of logarithms.

We start off with \(\log_{7} 7^2 - \log_{7} 7^3\). Based on the rule \(\log_{b} b^x = x\), each \(\log_{7} 7^x\) can be simplified to just \(x\). Hence, \(\log_{7} 7^2\) equals 2, and \(\log_{7} 7^3\) equals 3.

After simplifying, what began as a potentially daunting logarithm problem devolves into a simple subtraction: \(2 - 3\). This gives us the result of -1. Simplifying expressions in this way makes solving logarithmic problems far more approachable and less prone to errors. Always take it step by step, and leverage the properties of logarithms to break problems into smaller parts.