Problem 10 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-14,\) use \(\log 2 \approx 0.3010, \log 5 \approx 0.6990,\) and \(\log 7 \approx 0.8451\) to evaluate each logarithm without using a calculator. Then check your answer using a calculator. $$\log \frac{5}{7}$$

Step-by-Step Solution

Answer

The value of \(\log \frac{5}{7}\) is approximately -0.1461

Step 1: Understand the Quotient Rule

In logarithmic computations, the quotient rule is a useful tool. It states that \(\log\frac{a}{b} = \log a - \log b\). This rule has to be followed for this exercise.

Step 2: Apply the Quotient Rule

Applying the quotient rule to the given logarithm, we find that \(\log \frac{5}{7} = \log 5 - \log 7\).

Step 3: Substitute Values

Using provided values for \(\log 5\) (which is approximately 0.6990) and \(\log 7\) (which is approximately 0.8451), we substitute these into the equation. Therefore, \(\log \frac{5}{7} = 0.6990 - 0.8451\).

Step 4: Perform Subtraction

The subtraction results in -0.1461.

Key Concepts

Quotient Rule for Logarithms

Understanding the quotient rule for logarithms is pivotal for solving logarithmic equations where division is involved. This property essentially simplifies the process by breaking down complex expressions. The rule is straightforward: for any positive numbers a and b where both a and b do not equal 1, and for any base b, the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. Mathematically, it is expressed as
\[\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\].
By extracting and subtracting individual logarithms, this facilitates mental arithmetic or paper calculations without the immediate need for a calculator. Applying this rule to the given exercise,
\[\log \left(\frac{5}{7}\right) = \log(5) - \log(7)\]
leads to a simplified form that can be manually computed using familiar logarithm values.

Logarithmic Properties

Grasping the various logarithmic properties not only aids in computation but also enriches the conceptual understanding of logarithms. Beyond the quotient rule, there are other significant properties to know:

• Product Rule: For any positive numbers a and b, \(\log_b(ab) = \log_b(a) + \log_b(b)\).
• Power Rule: For any positive number a, \(\log_b(a^c) = c \cdot \log_b(a)\), where c is a real number.
• Change of Base Formula: For any positive numbers a and b, with b not equal to 1, you can change to a different base by using \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), where c is the new base you are converting to.
• Logarithm of 1: For any positive base b where b is not equal to 1, \(\log_b(1) = 0\).
• Logarithm of Base: For any positive base b, \(\log_b(b) = 1\).

These properties are fundamental in algebra and precalculus and can simplify complex logarithmic expressions. Logarithmic properties are often combined to solve various logarithmic equations, leveraging one or more properties in a stepwise manner to reach the solution.

Evaluating Logarithms Without a Calculator

In scenarios where a calculator is not available, understanding how to evaluate logarithms without a calculator becomes indispensable. It requires familiarity with common log values, the use of logarithmic properties, and sometimes estimation and pattern recognition. In the step-by-step solution presented in this exercise, known values of \(\log(2)\), \(\log(5)\), and \(\log(7)\) are employed to solve for \(\log\left(\frac{5}{7}\right)\). By substituting the given values – \(\log(5) \approx 0.6990\) and \(\log(7) \approx 0.8451\) – manual subtraction reveals the answer as -0.1461, showcasing a practical application of logarithmic evaluation.
Estimation plays a key role too, especially when dealing with logs to bases other than 10 or e (the natural logarithm). You can use patterns, like knowing that \(\log_b(b^n) = n\), to extrapolate unknown values from known ones. Also, recall that any log to the base of its number is 1, and the log of 1 to any base is 0. Integrating these simple tips can significantly ease the manual computation of logarithms in various contexts.