Problem 31 - Algebra and Trigonometry

Question

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{x} 16=4} & {\text { (b) } \log _{x} 8=\frac{3}{2}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(x = 2\); (b) \(x = 4\).

Step 1: Understand the logarithmic equation for part (a)

The equation given is \( \log_{x} 16 = 4 \). This means that the base \( x \) raised to the power of 4 equals 16 according to the definition of logarithms.

Step 2: Rewrite the logarithmic equation as an exponential equation for (a)

Using the definition of a logarithm, rewrite the equation \( \log_{x} 16 = 4 \) as an exponential equation: \( x^4 = 16 \).

Step 3: Solve for x in the equation for part (a)

To solve for \( x \), take the fourth root of both sides of the equation: \( x = 16^{\frac{1}{4}} \). Simplifying this gives \( x = 2 \).

Step 4: Understand the logarithmic equation for part (b)

The equation given is \( \log_{x} 8 = \frac{3}{2} \). This means that the base \( x \) raised to the power of \( \frac{3}{2} \) is equal to 8.

Step 5: Rewrite the logarithmic equation as an exponential equation for (b)

Using the definition of a logarithm, rewrite \( \log_{x} 8 = \frac{3}{2} \) as an exponential equation: \( x^{\frac{3}{2}} = 8 \).

Step 6: Solve for x in the equation for part (b)

To solve for \( x \), raise both sides to the power of \( \frac{2}{3} \): \( x = 8^{\frac{2}{3}} \). Simplifying this gives \( x = 4 \).

Key Concepts

Exponential Equation

An exponential equation is one where the variable appears in the exponent. This type of equation is essential in various real-world applications, such as population growth, radioactive decay, and cooling processes. Understanding these equations helps in determining unknowns when one has the base and the exponent. For example:

• In the equation \(x^4 = 16\), \(x\) is the base raised to the power of 4, equaling 16.
• In \(x^{\frac{3}{2}} = 8\), the exponent is a fraction, representing both a power and a root.

These equations often require the use of roots or logs to solve, providing a strong connection to logarithmic equations.

Definition of Logarithms

Logarithms are the inverse operations of exponentiation in mathematics. A logarithm answers the question: To what power must a certain base be raised to obtain another number? If we say \(\log_b a = c\), it translates to \(b^c = a\). Therefore, logarithms allow us to solve for unknown exponents.

• For instance, \(\log_{x} 16 = 4\) implies \(x^4 = 16\).
• Similarly, \(\log_{x} 8 = \frac{3}{2}\) implies \(x^{\frac{3}{2}} = 8\).

Understanding the definition and properties of logarithms enables us to convert logarithmic equations into exponential forms and solve them more easily.

Solving Logarithmic Equations

To solve logarithmic equations, we often turn them into exponential equations by utilizing the definition of logarithms. This step simplifies the process greatly. For example:

• Given \(\log_{x} 16 = 4\), we write it as \(x^4 = 16\) and solve by taking the fourth root: \(x = 16^{\frac{1}{4}} = 2\).
• For \(\log_{x} 8 = \frac{3}{2}\), it becomes \(x^{\frac{3}{2}} = 8\). To solve, raise both sides to the power of \(\frac{2}{3}\): \(x = 8^{\frac{2}{3}} = 4\).

By rewriting logarithms as exponentials, students can then apply roots or fractional exponents to find the unknown base, making it a straightforward process that demystifies the seemingly complex nature of logarithmic equations.