Problem 53 - Algebra A Combined Function

Question

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \frac{5}{3} $$

Step-by-Step Solution

Answer

\(\log_b \frac{5}{3} = 0.2\)

Step 1: Apply the Quotient Rule of Logarithms

Using the quotient rule for logarithms, which states that \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\), we can express \(\log_b \frac{5}{3}\) as follows: \(\log_b 5 - \log_b 3\).

Step 2: Substitute Given Values

Substitute the given values into the expression: \(\log_b 5 = 0.7\) and \(\log_b 3 = 0.5\). So, the expression becomes \(0.7 - 0.5\).

Step 3: Compute the Result

Perform the subtraction: \(0.7 - 0.5 = 0.2\). This is the value of \(\log_b \frac{5}{3}\).

Key Concepts

Quotient Rule

The Quotient Rule for logarithms is a powerful tool for simplifying and solving logarithmic expressions that involve division. It states:\[ \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \]This means that the logarithm of a quotient can be expressed as the difference of the logarithms of the numerator and the denominator. This rule makes it easier to calculate complicated logarithms without having to divide directly.For example, if you're asked to evaluate \( \log_b \frac{5}{3} \), the quotient rule allows you to transform it into \( \log_b 5 - \log_b 3 \), simplifying the computation process.
In mathematical operations, using the quotient rule saves time and reduces the chance of errors. It's like having a shortcut that helps you take large problems step by step.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases involving logarithms, which are the inverses of exponentials. They can be simplified using various rules, such as the quotient rule. They are used to solve equations where the unknowns appear as exponents.Logarithmic expressions can come in many forms, such as:

• Simple (e.g., \( \log_b x \))
• Complex (e.g., \( \log_b \left( x^2 + y^2 \right) \))
• Combined with operations (e.g., \( \log_b x + \log_b y \))
• Involving division (e.g., \( \log_b \frac{x}{y} \))

Simplifying these expressions often involves applying different properties and rules of logarithms to reduce them to a more manageable form. This makes it easier to plug in known values and perform calculations accurately.

Logarithmic Computation

Logarithmic computation involves using known values and properties of logarithms to evaluate a given logarithmic expression. For example, if you are given several logarithmic values, like \( \log_b 3 = 0.5 \) and \( \log_b 5 = 0.7 \), you can use these to compute other logarithmic values.In our example, we wanted to find \( \log_b \frac{5}{3} \). Here's how we go about it:

• First, apply the quotient rule to express the problem as \( \log_b 5 - \log_b 3 \).
• Then, substitute the given logarithmic values into the expression to get \( 0.7 - 0.5 \).
• Finally, compute the result by subtraction to get \( 0.2 \).

These steps simplify complex logarithmic expressions, making it easier for students to solve problems efficiently. Always remember that knowing the logarithmic properties is crucial for effective computation.