Problem 45 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} 96-\log _{2} 3 $$

Step-by-Step Solution

Answer

5

Step 1: Apply the Logarithm Quotient Rule

According to the quotient rule of logarithms, the difference of two logarithms with the same base can be rewritten as a single logarithm where the values inside the logarithms are divided. Hence, \[ \log_{2}96 - \log_{2}3 \] becomes \[ \log_{2}(96/3) \].

Step 2: Simplify the expression for calculation

Now, simplify the expression inside the logarithm. 96 divided by 3 is equal to 32. Therefore, the expression now becomes \[ \log_{2}(32) \].

Step 3: Evaluate the logarithmic expression

The expression \[ \log_{2}(32) \] is asking for the power to which 2 must be raised to get 32. We know that \(2^{5}\) is 32. Hence, the answer will be 5.

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a fundamental property used to simplify the subtraction of two logarithms that share the same base. This property states that:

• Whenever you subtract one logarithm from another, with the same base, you can express it as a single logarithm where the contents are divided, using the formula:
• \[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]

This concept can be incredibly helpful when trying to combine logarithmic expressions or solve more complex equations. In our specific example, by applying the quotient rule, we transform the expression:

• \(\log_2 96 - \log_2 3\)
• Into \(\log_2 (96/3)\)

This approach rapidly simplifies the problem, providing us with a clearer path to finding a solution.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves reducing the content of the logarithmic function to make evaluation easier. Our current expression, \(\log_2 (96/3)\), requires simplification of the fraction inside the logarithm.
The division of 96 by 3 results in the number 32, so the expression simplifies to:

• \(\log_2 32\)

Breaking down the problem step by step is crucial for simplifying complex expressions. Here, reducing 96 by 3 is straightforward, but each simplification step is important for clarity and ensuring precision. Simplification sets the stage for the next step, which often involves evaluating the result.

Evaluating Logarithms

Evaluating a logarithmic expression typically means finding the power to which the base must be raised to get a specific number. In our case, \(\log_2 32\) asks us to consider what power 2 requires to equal 32.
Upon recognizing powers of 2, we know that:

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)

Hence, 32 is 2 raised to the power of 5. Therefore, \(\log_2 32 = 5\). Understanding logarithm evaluations requires practice and familiarity with common powers. Developing this understanding allows you to quickly identify solutions to similar problems in the future.