Problem 16 - Precalculus Mathematics for Calculus

Question

Express the equation in logarithmic form. (a) \(4^{-3 / 2}=0.125\) (b) \(7^{3}=343\)

Step-by-Step Solution

Answer

(a) \(\log_4(0.125) = -\frac{3}{2}\); (b) \(\log_7(343) = 3\).

Step 1: Understand Logarithmic Form

To express an equation in logarithmic form, remember that if \(a^b = c\), it can be rewritten as \(\log_a(c) = b\). This form expresses the exponent as the result of a logarithm with base \(a\) and argument \(c\).

Step 2: Convert (a) to Logarithmic Form

The equation given is \(4^{-3/2} = 0.125\). Using the logarithmic form \(\log_a(c) = b\), identify \(a = 4\), \(b = -3/2\), and \(c = 0.125\). Rewrite this as \(\log_4(0.125) = -\frac{3}{2}\).

Step 3: Convert (b) to Logarithmic Form

The equation given is \(7^3 = 343\). Using the logarithmic form \(\log_a(c) = b\), identify \(a = 7\), \(b = 3\), and \(c = 343\). Rewrite this as \(\log_7(343) = 3\).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in mathematics, representing repeated multiplication of a base number. For instance, the expression \(a^b\) signifies that the base \(a\) is multiplied by itself \(b\) times. This notation helps simplify expressions and calculations by providing a concise way to represent large numbers.

• Base \(a\): The number that is being multiplied.
• Exponent \(b\): The power to which the base is raised, indicating how many times the multiplication occurs.
• For example, \(4^{-3/2}\) means \(4\) is raised to the power of \(-3/2\), which involves taking both a reciprocal and a root.

Exponents can be positive, negative, or even fractions. Negative exponents indicate division or reciprocals, while fractional exponents represent roots. Understanding these principles is crucial, especially when working with logarithms in precalculus, which are just another way of expressing these relationships.

Delving into Logarithms

Logarithms are the inverse operations of exponents. They answer the question: "To what power must a base number be raised, to produce a given value?" The logarithm \(\log_a(c) = b\) indicates that the base \(a\) must be raised to the power of \(b\) to yield \(c\).
Logarithmic functions are particularly useful for solving equations where the variable is an exponent. By converting exponential equations into logarithmic form, we can isolate and solve for the unknown exponent.

• Base \(a\): This is the same base used in an exponent.
• Argument \(c\): The result of the exponential function, i.e., the number obtained when the base is raised to the power.
• Result \(b\): The exponent to which the base is raised to reach the argument.

In the examples given:

• For \(4^{-3/2} = 0.125\), the base \(4\) raised to \(-3/2\) equals \(0.125\), so \(\log_4(0.125) = -3/2\).
• For \(7^3 = 343\), the base \(7\) raised to \(3\) equals \(343\), so \(\log_7(343) = 3\).

Precalculus and its Connection to Exponents and Logarithms

Precalculus serves as a bridge between algebra and calculus, preparing students for the complex concepts they will encounter in calculus. Within precalculus, a deep understanding of exponents and logarithms is vital as they form the foundation for exploring exponential growth, logarithmic scales, and more.

• Exponential Functions: These are functions where the variable is an exponent. They naturally model growth and decay, such as in population dynamics or radioactive decay.
• Logarithmic Functions: Often used in contexts such as measuring acidity (pH) or sound intensity (decibels), logarithms provide a way to manage large ranges of values efficiently.
• Problem-Solving: Mastery of converting between exponential and logarithmic forms is key to solving a variety of mathematical problems, from simple equations to complex real-world applications.

Precalculus encourages students to understand these concepts verbally, numerically, algebraically, and graphically. This holistic approach allows for flexible problem-solving techniques and deeper comprehension of mathematical relationships.