Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{x^{5} y^{3}}{3}$$

Step-by-Step Solution

Answer

\( \log_{2} \left( \frac{x^{5} y^{3}}{3} \right) = 5 \cdot \log_{2}(x) + 3 \cdot \log_{2}(y) - \log_{2}(3) \)

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). Apply this rule to the given expression: \[ \log_{2} \left( \frac{x^{5} y^{3}}{3} \right) = \log_{2}(x^{5} y^{3}) - \log_{2}(3) \].

Step 2: Apply the Product Rule of Logarithms

The product rule of logarithms is \( \log_b(MN) = \log_b(M) + \log_b(N) \). Use this on the expression \( \log_{2}(x^{5} y^{3}) \): \[ \log_{2}(x^{5} y^{3}) = \log_{2}(x^{5}) + \log_{2}(y^{3}) \].

Step 3: Apply the Power Rule of Logarithms

The power rule of logarithms is \( \log_b(M^n) = n \cdot \log_b(M) \). Apply this to \( \log_{2}(x^{5}) \) and \( \log_{2}(y^{3}) \): \[ \log_{2}(x^{5}) = 5 \cdot \log_{2}(x) \] and \( \log_{2}(y^{3}) = 3 \cdot \log_{2}(y) \).

Step 4: Combine all Steps

Substitute the results of the previous steps back into the expression: \[ \log_{2} \left( \frac{x^{5} y^{3}}{3} \right) = 5 \cdot \log_{2}(x) + 3 \cdot \log_{2}(y) - \log_{2}(3) \].

Key Concepts

Understanding the Quotient Rule

The quotient rule for logarithms is a useful tool when you are dealing with ratios or fractions in your logarithmic expressions. It states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. This rule is represented by the formula: \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). Consider an example: You have \( \log_2 \left( \frac{x^5 y^3}{3} \right) \). Using the quotient rule, you can split this into two separate logarithmic expressions:

• \( \log_{2}(x^{5} y^{3}) \) which corresponds to the numerator,
• \( \log_{2}(3) \) which corresponds to the denominator.

Thus, \( \log_2 \left( \frac{x^5 y^3}{3} \right) = \log_2(x^5 y^3) - \log_2(3) \). This transformation simplifies the problem and prepares it for any further manipulations using other logarithmic properties. By breaking the expression into smaller, manageable parts, each component can be addressed independently.

Demystifying the Product Rule

The product rule of logarithms comes into play when you have expressions that are the product of multiple terms inside a logarithm. It allows you to transform a complex multiplication inside a logarithm into an addition outside the logarithm. This is captured by the formula: \( \log_b(MN) = \log_b(M) + \log_b(N) \). Let's apply this rule to our expression \( \log_2(x^5 y^3) \). Here, the product is \( x^5 y^3 \), a multiplication of two separate parts:

• \( x^5 \)
• \( y^3 \)

The product rule allows us to write this as: \( \log_{2}(x^{5} y^{3}) = \log_2(x^5) + \log_2(y^3) \). By simplifying the multiplication into addition, the product rule helps make logarithmic expressions more straightforward to handle, especially when further equations or rules need to be applied.

Exploring the Power Rule

The power rule of logarithms is especially handy when the variable inside a logarithm is raised to an exponent. According to this rule, you can "bring down" the exponent as a multiplier in front of the logarithm. The formula is: \( \log_b(M^n) = n \cdot \log_b(M) \). For example, consider \( \log_2(x^5) \). With the power rule, you can express this as: \( 5 \cdot \log_2(x) \). This transformation simplifies handling the expression, especially when multiple logarithmic rules are in play. Similarly, applying the power rule to the term \( \log_2(y^3) \) results in: \( 3 \cdot \log_2(y) \). When used together with the quotient and product rules, the power rule aids in breaking down and simplifying complex logarithmic expressions, making them much easier to manage and solve.