Problem 101 - College Algebra

Question

Which one of the following is true? a. \(\frac{\log _{7} 49}{\log _{7} 7}=\log _{7} 49-\log _{7} 7\) b. \(\log _{b}\left(x^{3}+y^{3}\right)=3 \log _{b} x+3 \log _{b} y\) c. \(\log _{b}(x y)^{5}=\left(\log _{b} x+\log _{b} y\right)^{5}\) d. \(\ln \sqrt{2}=\frac{\ln 2}{2}\)

Step-by-Step Solution

Answer

Only option (d) is true. The other options (a), (b) and (c) are false.

Step 1: Analyzing option (a)

In option (a), apply the first logarithm rule i.e. \(\frac{\log _{b} a}{\log _{b} c}=log_{c} a\). This simplifies \(\frac{\log _{7} 49}{\log _{7} 7}\) to \(\log _{7} 49\). But \(\log _{7} 49-\log _{7} 7\) simplifies to \(2-1 = 1\), which is not equal to \(\log _{7} 49\) (2). Therefore, option (a) is false.

Step 2: Analyzing option (b)

From option (b), apply the power rule of logarithms, \(\log _{b} a^{n}=n \log _{b} a\), for \(x \). It is visible that the equation does not hold for \(y\), as \(y^{3}\) can not expand to \(3\log _{b} y\). Therefore, option (b) is false.

Step 3: Analyzing option (c)

In option (c), we apply the power rule of logarithms, on the left side to get \(5(\log _{b} x+ \log _{b} y)\). This does not match the right side of the equation. Therefore, option (c) is false.

Step 4: Analyzing option (d)

In option (d), apply the power rule of logarithms on the left side gives \(\frac{1}{2} \ln 2\), which matches the right side of the equation.

Key Concepts

Logarithm Properties

Logarithms have unique properties that make calculations easier by transforming multiplicative relations into additive ones. Learning these properties is vital to simplify complex expressions.

Here are some key properties:

• Product Rule: \( \log_b (xy) = \log_b x + \log_b y \) - helps in splitting the log of a product into the sum of individual logs.
• Quotient Rule: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \) - allows you to separate the log of a division.
• Power Rule: \( \log_b (x^n) = n \cdot \log_b x \) - makes it possible to bring exponents in front as a multiplier.
• Change of Base: \( \frac{\log_b a}{\log_b c} = \log_c a \) - converts a complex logarithm into one with a different base.

In the exercise, understanding these rules was essential to determine the validity of each statement. For example, the power and quotient rules were critical in evaluating the expressions.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). It's highly useful in calculus because it simplifies expressions involving \( e \), the base of natural exponential functions.

The properties of natural logarithms are similar to those of regular logarithms, with their own special use cases:

• Simplicity with growth rates: Often used in calculating continuous growth or decay, such as interest rates or population growth.
• Derivative formulas: In calculus, the derivative of \( \ln x \) is \( \frac{1}{x} \).
• Conversion and simplification: Natural log properties, like \( \ln \sqrt{2} = \frac{1}{2} \ln 2 \), make it easier to perform complex arithmetic.

In the provided exercise, recognizing that \( \ln \sqrt{2} \) matches \( \frac{\ln 2}{2} \) demonstrates the utility of the natural log in computations.

Logarithmic Equations

Logarithmic equations are equations in which the unknown appears inside a logarithm. Solving them involves applying logarithmic properties to simplify and rewrite equations.

• Example tasks: Simplifying complex expressions, transforming the equation into a solvable form.
• Use of inverse functions: The exponential and logarithm functions are inverses, used to isolate the variable.
• Application of rules: Utilizing the product, quotient, and power laws strategically helps in transforming and solving these equations.

When examining the exercise, testing each option against known logarithmic rules allowed for the identification of correct or incorrect formulations. Understanding these equations fosters a deeper insight into both the algebraic and real-world applications of logarithms.