Expanding Logarithmic Terms Using Rules of Logs

Introduction

Key Concepts

Understanding Logarithms

A logarithm is the inverse operation to exponentiation. It answers the question: to what exponent must a base be raised to produce a given number? Formally, for any positive real numbers \( a \), \( b \), and \( c \), where \( a \neq 1 \) and \( b \neq 1 \), $$ \log_b a = c \iff b^c = a. $$ This foundational definition allows us to manipulate and simplify complex logarithmic expressions using established rules.

Basic Properties of Logarithms

To expand logarithmic terms, it is crucial to understand the basic properties or rules of logarithms. These rules provide a systematic way to break down and simplify logarithmic expressions:

• Product Rule: The logarithm of a product is the sum of the logarithms of its factors. $$ \log_b (MN) = \log_b M + \log_b N $$
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms. $$ \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N $$
• Power Rule: The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. $$ \log_b (M^k) = k \cdot \log_b M $$
• Change of Base Formula: Allows the transformation of logarithms from one base to another. $$ \log_b a = \frac{\log_c a}{\log_c b} $$

Expanding Logarithmic Expressions

Expanding logarithmic terms involves applying the logarithm rules to rewrite complex logarithmic expressions into simpler, more manageable forms. This process is particularly useful in solving equations and simplifying expressions in algebra and calculus.

Example 1: Expanding a Product Inside a Logarithm

Consider the logarithmic expression \( \log_2 (8xy^2) \). Using the product rule: $$ \log_2 (8xy^2) = \log_2 8 + \log_2 x + \log_2 y^2 $$ Applying the power rule to \( \log_2 y^2 \): $$ \log_2 (8xy^2) = \log_2 8 + \log_2 x + 2 \cdot \log_2 y $$ Since \( 8 = 2^3 \), \( \log_2 8 = 3 \): $$ \log_2 (8xy^2) = 3 + \log_2 x + 2 \log_2 y $$>

Example 2: Expanding a Quotient Inside a Logarithm

Take \( \log_3 \left( \frac{27m^4}{n} \right) \). Applying the quotient rule: $$ \log_3 \left( \frac{27m^4}{n} \right) = \log_3 27m^4 - \log_3 n $$ Now, expand \( \log_3 27m^4 \) using the product rule: $$ \log_3 27m^4 = \log_3 27 + \log_3 m^4 $$ And apply the power rule to \( \log_3 m^4 \): $$ \log_3 27m^4 = \log_3 27 + 4 \cdot \log_3 m $$ Since \( 27 = 3^3 \), \( \log_3 27 = 3 \): $$ \log_3 \left( \frac{27m^4}{n} \right) = 3 + 4 \log_3 m - \log_3 n $$>

Combining Logarithmic Rules for Complex Expansions

Often, logarithmic expressions require the application of multiple rules to fully expand them. The strategy involves identifying the structure of the expression and systematically applying the appropriate rules.

Example 3: Expanding a Logarithm with Nested Products and Quotients

Expand \( \log_5 \left( \frac{50x^3y^2}{z^4} \right) \). Step 1: Apply the quotient rule: $$ \log_5 \left( \frac{50x^3y^2}{z^4} \right) = \log_5 (50x^3y^2) - \log_5 z^4 $$ Step 2: Expand \( \log_5 (50x^3y^2) \) using the product rule: $$ \log_5 (50x^3y^2) = \log_5 50 + \log_5 x^3 + \log_5 y^2 $$ Step 3: Apply the power rule: $$ \log_5 50 + 3 \log_5 x + 2 \log_5 y $$ Step 4: Expand \( \log_5 z^4 \) using the power rule: $$ 4 \log_5 z $$ Final Expanded Form: $$ \log_5 \left( \frac{50x^3y^2}{z^4} \right) = \log_5 50 + 3 \log_5 x + 2 \log_5 y - 4 \log_5 z $$

Applications of Expanding Logarithmic Terms

Expanding logarithmic terms is not only a theoretical exercise but also has practical applications in various fields such as engineering, computer science, and economics. For instance:

• Solving Exponential Equations: By expanding logarithmic expressions, we can isolate variables and solve for unknowns in equations involving exponents.
• Data Analysis: Logarithms are used to transform multiplicative relationships into additive ones, simplifying the analysis of data with exponential growth or decay.
• Complexity Analysis: In computer science, logarithms help analyze the efficiency of algorithms, especially those with logarithmic time complexities.

Common Mistakes and How to Avoid Them

When expanding logarithmic terms, students often encounter pitfalls that can lead to incorrect results. Being aware of these common mistakes can enhance accuracy and understanding:

• Misapplying the Product and Quotient Rules: Ensure that you correctly identify the product and quotient within the logarithmic argument before applying the respective rules.
• Incorrect Use of the Power Rule: Remember that the power rule applies only to the exponent outside the logarithm. Misapplying it to coefficients or other parts of the expression can lead to errors.
• Neglecting the Domain of Logarithms: Logarithmic functions are defined only for positive real numbers. Ensure that all arguments within the logarithms remain positive after expansion.
• Forgetting to Simplify Constants: After expansion, constants can often be simplified further, especially if they are powers of the base. Always check if constants can be reduced to simpler forms.

Practice Problems

1. Expand the logarithmic expression \( \log_4 (16a^2b) \).
2. Simplify \( \log_{10} \left( \frac{1000x^3}{y^2} \right) \).
3. Given \( \log_2 \left( \frac{8m^4n}{p^5} \right) \), expand it using the rules of logarithms.
4. Expand \( \log_7 \left( 49x \sqrt{y} \right) \).
5. Simplify \( \log_5 \left( \frac{125a^3}{b^2c} \right) \).

Answers:

1. \( \log_4 (16a^2b) = \log_4 16 + \log_4 a^2 + \log_4 b = 2 + 2 \log_4 a + \log_4 b \)
2. \( \log_{10} \left( \frac{1000x^3}{y^2} \right) = \log_{10} 1000 + 3 \log_{10} x - 2 \log_{10} y = 3 + 3 \log_{10} x - 2 \log_{10} y \)
3. \( \log_2 \left( \frac{8m^4n}{p^5} \right) = \log_2 8 + 4 \log_2 m + \log_2 n - 5 \log_2 p = 3 + 4 \log_2 m + \log_2 n - 5 \log_2 p \)
4. \( \log_7 \left( 49x \sqrt{y} \right) = \log_7 49 + \log_7 x + \log_7 y^{1/2} = 2 + \log_7 x + \frac{1}{2} \log_7 y \)
5. \( \log_5 \left( \frac{125a^3}{b^2c} \right) = \log_5 125 + 3 \log_5 a - 2 \log_5 b - \log_5 c = 3 + 3 \log_5 a - 2 \log_5 b - \log_5 c \)

Advanced Techniques: Combining Multiple Log Rules

In more complex scenarios, expanding logarithmic expressions may require a combination of multiple logarithmic rules. Mastery of these techniques enables students to tackle advanced problems with confidence.

Example 4: Expanding a Logarithm with Multiple Operations

Expand \( \log_3 \left( \frac{81x^6}{y^3 \sqrt{z}} \right) \). Step 1: Apply the quotient rule: $$ \log_3 \left( \frac{81x^6}{y^3 \sqrt{z}} \right) = \log_3 (81x^6) - \log_3 (y^3 \sqrt{z}) $$>

Step 2: Expand \( \log_3 (81x^6) \) using the product rule: $$ \log_3 81 + \log_3 x^6 $$ Apply the power rule: $$ 4 + 6 \log_3 x $$ Step 3: Expand \( \log_3 (y^3 \sqrt{z}) \) using the product rule: $$ \log_3 y^3 + \log_3 z^{1/2} $$>

Apply the power rule: $$ 3 \log_3 y + \frac{1}{2} \log_3 z $$>

Step 4: Combine the results: $$ \log_3 \left( \frac{81x^6}{y^3 \sqrt{z}} \right) = 4 + 6 \log_3 x - 3 \log_3 y - \frac{1}{2} \log_3 z $$

Logarithmic Equations Involving Expanded Terms

Once logarithmic expressions are expanded, they can be used to solve more complex equations. This often involves isolating the logarithmic terms or converting them back into exponential form.

Example 5: Solving an Equation with Expanded Logarithmic Terms

Solve for \( x \) in the equation: $$ 2 \log_2 x + \log_2 8 = 5 $$ Step 1: Apply the power rule to the first term: $$ 2 \log_2 x = \log_2 x^2 $$>

Step 2: Combine the logarithmic terms using the product rule: $$ \log_2 x^2 + \log_2 8 = \log_2 (8x^2) $$>

So, the equation becomes: $$ \log_2 (8x^2) = 5 $$>

Step 3: Convert the logarithmic equation to its exponential form: $$ 8x^2 = 2^5 = 32 $$>

Step 4: Solve for \( x^2 \): $$ x^2 = \frac{32}{8} = 4 $$>

Step 5: Take the square root of both sides: $$ x = \pm 2 $$>

However, since logarithms of negative numbers are undefined, the solution is: $$ x = 2 $$>

Using Technology to Verify Expanded Logarithmic Expressions

Graphing calculators and computer algebra systems (CAS) can be invaluable tools for verifying the correctness of expanded logarithmic expressions. By inputting both the original and the expanded forms into a calculator, students can graph the functions or evaluate them at specific points to ensure they are equivalent.

For example, to verify \( \log_2 (8xy^2) = 3 + \log_2 x + 2 \log_2 y \), choose values for \( x \) and \( y \), compute both sides, and confirm that they match.

Real-World Applications: Modeling Growth and Decay

Logarithmic expansions are instrumental in modeling exponential growth and decay, such as population dynamics, radioactive decay, and financial calculations involving compound interest. By expanding logarithmic terms, one can linearize exponential models, making them easier to analyze and interpret.

For instance, in population growth, the number of individuals can be expressed as \( P(t) = P_0 e^{kt} \). Taking the natural logarithm of both sides: $$ \ln P(t) = \ln P_0 + kt $$>

Here, the logarithmic expansion simplifies the model, allowing for linear regression analysis to determine growth rates.

Advanced Practice: Expanding Nested Logarithmic Expressions

Some logarithmic expressions involve nesting, where one logarithm is inside another. Expanding such expressions requires careful application of the logarithm rules in a step-by-step manner.

Example 6: Expanding a Nested Logarithm

Expand \( \log_2 (\log_4 x^2) \). Step 1: Simplify the inner logarithm using the change of base formula: $$ \log_4 x^2 = \frac{\log_2 x^2}{\log_2 4} = \frac{2 \log_2 x}{2} = \log_2 x $$>

Step 2: Substitute back into the original expression: $$ \log_2 (\log_4 x^2) = \log_2 (\log_2 x) $$>

Thus, \( \log_2 (\log_4 x^2) = \log_2 (\log_2 x) \), demonstrating the simplification of a nested logarithmic expression.

Comparison Table

Summary and Key Takeaways