Problem 72 - Algebra for College Students

Question

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{x^{2}}{y}\right) $$

Step-by-Step Solution

Answer

\( 2 \cdot \log_b(x) - \log_b(y) \)

Step 1: Rewrite as a difference

Using the quotient rule of logarithms, which states \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \), we separate the expression: \( \log_b\left(\frac{x^2}{y}\right) \) becomes \( \log_b(x^2) - \log_b(y) \).

Step 2: Apply the power rule

The power rule of logarithms states that \( \log_b(M^n) = n \cdot \log_b(M) \). Applying this to \( \log_b(x^2) \), we get \( 2 \cdot \log_b(x) \).

Step 3: Final Expression

Incorporating both steps, the original expression \( \log_b\left(\frac{x^2}{y}\right) \) simplifies to \( 2 \cdot \log_b(x) - \log_b(y) \).

Key Concepts

Quotient Rule of Logarithms

Logarithms are a powerful tool for simplifying complex mathematical relationships. When dealing with division inside a logarithmic expression, the quotient rule of logarithms comes to the rescue. This rule states that for any positive real numbers \( M \) and \( N \), and a positive base \( b \), the logarithm of a quotient is the difference of the logarithms:

• \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)

This means if you're dividing two numbers inside a log, you can separate them into two distinct logs with a subtraction sign in between.
For example, in the expression \( \log_b\left(\frac{x^2}{y}\right) \), we can apply the quotient rule to break it down into two simpler parts: \( \log_b(x^2) - \log_b(y) \).
Breaking expressions down in this way makes them easier to work with, especially when further simplifications are necessary. Understanding and using the quotient rule can greatly streamline the process of solving logarithmic equations.

Power Rule of Logarithms

The power rule of logarithms is another handy tool in simplifying expressions, especially when dealing with exponents. This identity shows how to handle logarithms where the inside number is raised to a power. The rule is:

• \( \log_b(M^n) = n \cdot \log_b(M) \)

This means that when a number \( M \) inside a log is raised to a power \( n \), you can "bring down" the exponent and multiply it by the log of the base number.
In the context of our example, \( \log_b(x^2) \) can be simplified using the power rule, becoming \( 2 \cdot \log_b(x) \).
Utilizing the power rule can make complex exponential expressions more straightforward and manageable. Remember that the power comes out to the front as a coefficient before the logarithm.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves applying multiple logarithmic identities such as the quotient and power rules in combination. This process helps transform a complicated logarithmic statement into something easier to understand and solve.Let’s consider the full simplification of the expression \( \log_b\left(\frac{x^2}{y}\right) \). First, apply the quotient rule to separate the log expression into a difference: \( \log_b(x^2) - \log_b(y) \).
Next, further simplify each logarithmic component using the power rule. Specifically, \( \log_b(x^2) \) becomes \( 2 \cdot \log_b(x) \). This allows the expression to become: \( 2 \cdot \log_b(x) - \log_b(y) \).
By employing these strategic simplifications, calculations become more straightforward and less prone to error. Simplifying carefully and systematically thus makes it easier to work with logarithmic expressions in practical scenarios.