Problem 26 - Calculus and its applications

Question

Given \(\log _{b} 3=1.099\) and \(\log _{b} 5=1.609\), find each value. $$ \log _{b} \frac{5}{3} $$

Step-by-Step Solution

Answer

The value of \( \log_b \frac{5}{3} \) is 0.51.

Step 1: Apply the Quotient Rule for Logarithms

Recall the quotient rule for logarithms which states that \( \log_b \frac{x}{y} = \log_b x - \log_b y \). This allows us to separate the logarithm of a quotient into the difference of two logarithms.

Step 2: Substitute the Known Values

Using the values given in the problem, substitute \( \log_b 3 = 1.099 \) and \( \log_b 5 = 1.609 \) into the equation from Step 1. This gives us:\[ \log_b \frac{5}{3} = \log_b 5 - \log_b 3 = 1.609 - 1.099 \]

Step 3: Perform the Subtraction

Calculate the value of \( 1.609 - 1.099 \). The result is:\[ 1.609 - 1.099 = 0.51 \]

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a handy tool in simplifying logarithmic expressions that involve division. When you have a logarithm of a quotient, such as

• \( \log_b \frac{x}{y} \)

this rule allows you to break it down into the difference between two separate logarithms:

• \( \log_b x - \log_b y \)

With this rule, you decompose a complex logarithm into simpler parts, facilitating an easier calculation. In the given exercise, it helps convert the problematic expression \( \log_b \frac{5}{3} \) into more manageable pieces: \( \log_b 5 - \log_b 3 \). By transforming the original expression, you can substitute known values directly, making it straightforward to compute the final result.

Logarithmic Subtraction

Once you've applied the quotient rule for logarithms, you find yourself with an expression involving subtraction. This is referred to as logarithmic subtraction, where the task is to subtract one logarithm from another.

• This process hinges on knowing the values of individual logarithmic expressions, like \( \log_b 5 \) and \( \log_b 3 \).

In the example, these values are provided: \( \log_b 5 = 1.609 \) and \( \log_b 3 = 1.099 \).To carry out logarithmic subtraction, you simply perform the subtraction operation as you would with normal numbers:

• \( 1.609 - 1.099 \)

The subtraction within the context of logarithms transforms the relationship between the numbers (in this case 5 and 3) into a difference of exponential magnitudes, giving you the logarithm of the original division.

Logarithmic Calculation

After applying the quotient rule for logarithms and performing logarithmic subtraction, you reach the final step: logarithmic calculation. This involves calculating the numerical result of subtracting the logarithmic values. In mathematical terms, it translates to:

• \( 1.609 - 1.099 = 0.51 \)

This step might seem trivial, but it's crucial as it yields the precise logarithmic measure of the original expression's quotient: \( \log_b \frac{5}{3} \).Logarithmic calculation involves simple arithmetic here, but it underscores a significant transformation: turning a compound logarithmic problem into straightforward arithmetic. This reinforces the power of logarithmic rules to simplify what might initially seem like complex expressions.

• The final result, 0.51, is the concise logarithmic representation of dividing 5 by 3 in the given base \( b \).