Problem 12 - Technical Mathematics with Calculus

Question

Find the value of \(x\) in each expression. $$\log _{x} 16=4$$

Step-by-Step Solution

Answer

\(x = 2\)

Step 1: Understand the Logarithmic Equation

The given equation is in the form of a logarithm, \(\log_{x} 16 = 4\). It states that \(x\) raised to the power of 4 equals 16. To solve for \(x\), we need to convert the logarithmic equation into its exponential form.

Step 2: Convert to Exponential Form

The exponential form of the equation \(\log_{x} 16 = 4\) is \(x^4 = 16\). This is obtained by understanding that \(\log_{b} a = c\) is equivalent to \(b^c = a\), where \(b\) is the base of the logarithm, \(a\) is the result, and \(c\) is the exponent.

Step 3: Solve for \(x\)

To find \(x\), we solve the equation \(x^4 = 16\). Since 16 is a perfect square, specifically \(2^4\), we can deduce that \(x = 2\), because \(2^4 = 16\).

Key Concepts

Logarithm Properties

Logarithms are mathematical tools that allow us to solve equations involving exponents in a more manageable way. One of the fundamental properties of logarithms that is of paramount importance is the definition itself: If we have an equation \( \log_{b} a = c \), this tells us that the base \(b\) raised to the power \(c\) will give the value \(a\). This property is essential as it forms the bridge between logarithmic and exponential forms, allowing us to translate back and forth as needed for solving equations.

Other properties of logarithms that come into play with more complex equations include the Product Rule \(\log_b(mn) = \log_b(m) + \log_b(n)\), the Quotient Rule \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\), and the Power Rule \(\log_b(m^p) = p\log_b(m)\). Recognizing these properties can help simplify logarithmic expressions, making it easier to find the solution to the equation at hand.

Exponential Form Conversion

Conversion between logarithmic and exponential forms is a routine strategy in mathematics that aids in solving equations involving logarithms. To convert a logarithmic equation like \(\log_{x} 16 = 4\) into its exponential equivalent, we apply the fundamental definition of a logarithm. As outlined in the solution, \(x^4 = 16\) is the exponential form where \(x\) is the base, 4 is the exponent, and 16 is the result.

This conversion is not just a mechanical process but a conceptual understanding that opens the door to exponential equations that are often more straightforward to solve. In practice, to convert \(\log_{b} a = c\) to its exponential form, we simply write \(b^c = a\). This method allows students to leverage their knowledge of powers and roots to find the unknown variable. Additionally, if the equation has a known base and result, students can use the inverse operations associated with the base to find the exponent directly.

Mathematical Problem Solving

Mathematical problem solving encompasses a systematic approach to understanding and resolving mathematical challenges. The process often starts with comprehending the equation or problem statement, as was the case with identifying that \(\log_{x} 16 = 4\) is a logarithmic equation. Once the problem's nature is understood, converting complex expressions into simpler, more solvable forms, such as exponential equations, becomes the next step. This is enabled by using known mathematical properties and operations.

After obtaining a solvable equation, we apply algebraic methods to find the unknown variable. In the given example, recognizing that 16 is \(2^4\) is a crucial problem-solving insight that aligns with the general strategy of looking for patterns and known quantities to simplify and solve the equation. A systematic approach combined with critical thinking and pattern recognition is the crux of effective mathematical problem solving and paves the way for success in more complex algebraic challenges.