Problem 45 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log _{2} 96-\log _{2} 3 $$

Step-by-Step Solution

Answer

The expression \(\log _{2} 96 - \log _{2} 3\) simplifies to \(5\).

Step 1: Identify appropriate logarithmic property

Properties of logarithms can simplify the calculation. We focus on the property \(\log_b a - \log_b c = \log_b \frac{a}{c}\), because the exercise involves subtraction between two logarithms.

Step 2: Applying logarithmic property

Using the property identified in Step 1, the expression \(\log _{2} 96 - \log _{2} 3\) simplifies to \(\log _{2} \frac{96}{3}\).

Step 3: Calculation

Now, simplifying the fraction gives us the single logarithm expression \(\log _{2} 32\).

Step 4: Evaluate logarithmic expression

\(\log _{2} 32\) represents the power to which '2' must be raised to obtain 32. Since \(2^5 = 32\), the value of the expression \(\log _{2} 32\) is \(5\).

Key Concepts

Understanding Exponential Functions

An exponential function is a mathematical expression of the form \( f(x) = a \cdot b^x \), where \(a\) is a constant, \(b\) is the base of the exponential, and \(x\) is the exponent. These functions are fundamental in describing growth or decay processes, such as population growth, radioactive decay, and interest calculations.
The base \(b\) of an exponential function is typically greater than 0, and not equal to 1, leading to two distinct behaviors:

• If \(b > 1\), the function models growth.
• If \(0 < b < 1\), the function models decay.

In the context of logarithms, exponential functions help us understand the inverse operations. For example, if we know that \(b^y = x\), taking the logarithm of both sides gives \(y = \log_b x\). Understanding this relationship allows us to evaluate and manipulate logarithmic expressions effectively.

Condensing Logarithms Made Simple

Condensing logarithms involves using logarithmic properties to combine multiple logarithmic terms into a single one. This process simplifies complex expressions, making them more manageable and often easier to solve.
One of the key properties useful for condensing is the Quotient Rule, applicable when you have a subtraction between two logs with the same base:

• \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \)

In the exercise example, we used this property to condense \(\log_2 96 - \log_2 3\) into \(\log_2 \frac{96}{3}\). The goal is to transform multiple terms into one using mathematical simplification.
This property is incredibly useful for solving logarithmic equations and simplifying expressions where combining terms leads to solving for a single unknown or simplifying a problem to its core components.

Evaluating Logarithms Easily

Evaluating logarithms means calculating the value that an exponential base must be raised to obtain a certain number. This process is crucial for interpreting the results of condensed logarithms.
For instance, if you have \(\log_2 32\), it asks, "To what power must 2 be raised to result in 32?" Knowing that \(2^5 = 32\), we see that \(\log_2 32 = 5\).
Here are some handy principles for evaluating logarithms:

• If you have \(\log_b b = 1\): any number to the first power is itself.
• Zero logs, \(\log_b 1 = 0\): any number raised to the power of zero is 1.
• Powers of the base, directly evaluate when the number is a known power of the base.

Evaluating logarithms helps demystify and give concrete answers to what the expressions signify. This skill translates into solving larger equations and understanding relationships in exponential models.