Further Logarithms

Introduction

In this topic, we will delve into further logarithms, building upon the foundational knowledge of logarithms. Logarithms are a crucial mathematical concept that aids in simplifying complex calculations and solving exponential equations. We will explore advanced properties and applications of logarithms to enhance our problem-solving skills.

Product Rule

The product rule states that $\log_ab + \log_ac = \log_a(bc)$. This rule allows us to combine the sum of two logarithms into a single logarithm.

Example:
Simplify the expression: $\log_2 8 + \log_2 16$.

Solution:
Using the product rule, we have:
$\log_2 8 + \log_2 16 = \log_2 (8 \times 16) = \log_2 128$

Quotient Rule

The quotient rule states that $\log_ab - \log_ac = \log_a\left(\frac{b}{c}\right)$. This rule enables us to subtract one logarithm from another logarithm.

Example:
Evaluate: $\log_3 81 - \log_3 9$.

Solution:
Applying the quotient rule, we get:
$\log_3 81 - \log_3 9 = \log_3 \left(\frac{81}{9}\right) = \log_3 9$

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another. It states that $\log_ab = \frac{\log_cb}{\log_ca}$.

Example:
Convert $\log_3 27$ to a base 2 logarithm.

Solution:
Using the change of base formula, we have:
$\log_3 27 = \frac{\log_2 27}{\log_2 3}$

Power Rule

The power rule states that $n\log_ab = \log_a(b^n)$. This rule simplifies logarithms involving exponents.

Example:
Simplify: $5\log_2 4$.

Solution:
Applying the power rule, we get:
$5\log_2 4 = \log_2 4^5 = \log_2 1024$

Common Mistakes

1. Forgetting to apply the appropriate logarithmic rule when simplifying expressions.
2. Misusing the properties of logarithms, leading to incorrect solutions.
3. Not converting logarithms to a common base when necessary.

Key Points

• Product Rule: $\log_ab + \log_ac = \log_a(bc)$
• Quotient Rule: $\log_ab - \log_ac = \log_a\left(\frac{b}{c}\right)$
• Change of Base Formula: $\log_ab = \frac{\log_cb}{\log_ca}$
• Power Rule: $n\log_ab = \log_a(b^n)$

Practice Questions

1. Evaluate $\log_5 125 - \log_5 5$.

Solution:
$\log_5 125 - \log_5 5 = \log_5 \left(\frac{125}{5}\right) = \log_5 25$

2. Convert $\log_7 49$ to a base 3 logarithm.

Solution:
Using the change of base formula:
$\log_7 49 = \frac{\log_3 49}{\log_3 7}$

3. Simplify $3\log_4 16$.

Solution:
Using the power rule:
$3\log_4 16 = \log_4 16^3 = \log_4 4096$

4. If $\log_2 x = 5$, find the value of $x$.

Solution:
Since $\log_2 x = 5$, we have:
$2^5 = x$
$x = 32$

5. Evaluate $\log_6 216 - \log_6 6$.

Solution:
$\log_6 216 - \log_6 6 = \log_6 \left(\frac{216}{6}\right) = \log_6 36$