Problem 99 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ \text { If } f(u)=2 f(v), \text { then } v=u^{2} $$

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the Given Function

The given function \(f(x) = \ln x\) is a logarithmic function. Logarithmic functions have the property that \( \ln(a*b) = \ln(a) + \ln(b)\) and \(\ln(a/b) = \ln(a) - \ln(b)\).

Step 2: Understand the Statement to be Evaluated

We are given that \(f(u)=2 f(v)\), in other words \( \ln (u) = 2 * \ln (v) \). This can be rewritten as \( \ln (u) = \ln (v^2) \), using the properties of the logarithm.

Step 3: Evaluate the Equality of the Statement

From the equation \( \ln (u) = \ln (v^2) \), we know the argument of the logarithms on either side of the equation must be the same, therefore \(u = v^2\). Thus the statement is true.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial for working with logarithmic functions. A logarithm is the inverse operation to exponentiation, meaning the logarithm of a number is the exponent to which a base, typically e (Euler's number) for natural logarithms, must be raised to obtain that number. The basics properties of logarithms include:

• The product rule: \( \ln(a \times b) = \ln(a) + \ln(b) \).
• The quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• The power rule: \( \ln(a^p) = p \times \ln(a) \), where \(p\) is any real number.

These properties are incredibly useful when simplifying and solving logarithmic equations. By rewriting the log of a product, quotient, or power, you can transform complex logarithmic expressions into more manageable pieces.

Algebraic Problem Solving

Algebraic problem solving involves finding unknown variables or simplifying expressions using algebraic rules and properties. When it comes to logarithmic functions, algebra can help us manipulate expressions to isolate the unknown or to compare two expressions. To be proficient in algebraic problem solving, one should:

• Understand the properties of algebraic operations.
• Be skilled at applying laws of exponents and logarithms.
• Know how to isolate variables and change the subject of a formula.

These competencies allow us to transform complex equations into simpler forms that are easier to interpret or solve. Additionally, recognizing patterns and familiar structures in equations can help to apply the correct algebraic strategies.

Logarithmic Equations

Logarithmic equations contain logarithmic expressions and require the properties of logarithms for their solutions. Solving logarithmic equations usually involves using logarithmic properties to combine or break apart logarithmic terms. The process typically includes these steps:

• Apply the logarithmic properties to simplify the equation.
• Use algebraic techniques to isolate the logarithmic term if there's more than one term.
• Convert the logarithmic equation into an exponential form if necessary, as logarithms and exponents are inverse operations.
• Solve for the unknown, which might require further algebraic manipulation after removing the logarithms.

An important consideration when solving logarithmic equations is to check that solutions are within the domain of the logarithm, which for natural logarithms is positive real numbers only. Any 'solution' that isn't in the domain must be discarded as extraneous.