Algebra and Trigonometry: Properties of Logarithms

Properties of Logarithms and Core Identities

• Foundational Definitions and Domains:   - For all properties established in this section, $M$ and $a$ must be positive real numbers.   - The base $a$ must satisfy the condition $a \neq 1$.   - The variable $r$ represents any real number.

• Theorem: Basic Properties of Logarithms (1 of 4):   - The Exponentiation Property: The number $\log_a(M)$ is the specific exponent to which the base $a$ must be raised to obtain the value $M$.     - Mathematically expressed as: $a^{\log_a(M)} = M$   - The Inverse Power Property: The logarithm with base $a$ of the base $a$ raised to a specific power is equal to that power.     - Mathematically expressed as: $\log_a(a^r) = r$

• Establishing Fundamental Logarithmic Values (Example 1):   - Proving $\log_a(1) = 0$:     - This identity was originally established through graphing, but can be proven algebraically.     - Let $y = \log_a(1)$.     - Convert to exponential form: $a^y = 1$.     - Since any base raised to the power of $0$ is $1$, we solve for $y$ to find $y = 0$.     - Conclusion: $\log_a(1) = 0$   - Proving $\log_a(a) = 1$:     - Let $y = \log_a(a)$.     - Convert to exponential form: $a^y = a$.     - Since $a^1 = a$, we solve for $y$ to find $y = 1$.     - Conclusion: $\log_a(a) = 1$

• Examples of Basic Property Application (Example 2):   - Scenario A: $\log_{0.2}(0.2)^{-\sqrt{2}}$ results in $-\sqrt{2}$ by applying $\log_a(a^r) = r$.   - Scenario B: $\ln(e^{kt})$ results in $kt$ (noting that the base of the natural log is $e$).   - Scenario C: $2^{\log_2(\pi)}$ results in $\pi$ by applying $a^{\log_a(M)} = M$.

Expansion and Condensation of Logarithmic Expressions

• Theorem: Operational Properties of Logarithms (2 of 4):   - The Product Rule: The log of a product is equal to the sum of the logs of its factors.     - $\log_a(MN) = \log_a(M) + \log_a(N)$   - The Quotient Rule: The log of a quotient is equal to the difference of the logs of the dividend and divisor.     - $\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$

• Theorem: Power Property of Logarithms (3 of 4):   - The Power Rule: The log of a value raised to a power is equal to the product of that power and the log of the value.     - $\log_a(M^r) = r\log_a(M)$

• Expansion Examples (Writing as a Sum or Difference):   - Example 3: Expanding $\log_a(x\sqrt{x^2+1})$ as a sum of logarithms.     - By the Product Rule: $\log_a(x) + \log_a(\sqrt{x^2+1})$     - Expressing the radical as a power: $\log_a(x) + \log_a((x^2+1)^{\frac{1}{2}})$     - Using the Power Rule to express exponents as factors: $\log_a(x) + \frac{1}{2}\log_a(x^2+1)$   - Example 4: Expanding a quotient expression as a difference of logarithms while expressing all powers as factors.   - Example 5: Expanding a complex expression involving products, quotients, and square roots into a sum and difference of logarithms, ensuring all powers are moved to the front as factors.

• Condensation Examples (Example 6: Writing as a Single Logarithm):   - Scenario A: $\log_a(7) + \log_a(3) = \log_a(7 \times 3) = \log_a(21)$.   - Scenario B: $2\log_a(x) + \frac{1}{2}\log_a(9)$.     - Apply the Power Rule in reverse: $\log_a(x^2) + \log_a(9^{\frac{1}{2}})$.     - Simplify the constant: $\log_a(x^2) + \log_a(3)$.     - Apply the Product Rule: $\log_a(3x^2)$.   - Scenario C: $\log_a(x+1) - \log_a(x) = \log_a\left(\frac{x+1}{x}\right)$.

Equality and Approximation Techniques

• Theorem: Logarithmic Equality (4 of 4):   - If $M = N$, then $\log_a(M) = \log_a(N)$.   - Conversely, if $\log_a(M) = \log_a(N)$, then $M = N$.   - This assumes $M$, $N$, and $a$ are positive real numbers and $a \neq 1$.

• Approximation without Common Bases (Example 7):   - Task: Approximate $\log_4(17)$ and round to four decimal places.   - Conceptual Logic: The expression $\log_4(17)$ asks: "4 raised to what exponent equals 17?"   - Let $y = \log_4(17)$.   - Estimation:     - $4^2 = 16$     - $4^3 = 64$     - Therefore, the value of $y$ must be between $2$ and $3$, and significantly closer to $2$ than to $3$.   - Numerical Result: Rounded to four decimal places, the value is approximately $2.0437$.

The Change-of-Base Formula

• Theorem: Change-of-Base Formula:   - This formula allows for the conversion of a logarithm with any valid base $a$ into a ratio of logarithms with a new base $b$.   - If $a \neq 1$, $b \neq 1$, and $M$ are positive real numbers, then: $\log_a(M) = \frac{\log_b(M)}{\log_b(a)}$   - In practice, the formula is most useful when converting to base $10$ (common logarithm) or base $e$ (natural logarithm) because these are standard on calculators:     - Base 10: $\log_a(M) = \frac{\log(M)}{\log(a)}$     - Base e: $\log_a(M) = \frac{\ln(M)}{\ln(a)}$

• Application of Change-of-Base (Example 8):   - Scenario A: Approximate $\log_5(89)$.     - Calculation: $\frac{\ln(89)}{\ln(5)} \approx 2.7889$   - Scenario B: Approximate $\log_{\sqrt{2}}(\sqrt{5})$.     - Calculation: $\frac{\log(\sqrt{5})}{\log(\sqrt{2})} \approx 2.3219$

Graphing with Non-Standard Bases (Example 9)

• Procedure for Graphing Utility:   - Graphing utilities typically only have keys for $\log_{10}$ ($\log$) and $\log_e$ ($\ln$).   - To graph a function like $y = \log_2(x)$, one must use the Change-of-Base Formula.   - Step 1: Express the function in terms of base $10$ or base $e$.     - Conversion: $y = \frac{\log(x)}{\log(2)}$ or $y = \frac{\ln(x)}{\ln(2)}$.   - Step 2: Input either of these quotient expressions into the graphing utility to render the graph of $y = \log_2(x)$.