Problem 12 - Algebra and Trigonometry

Question

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 4^{-3 / 2}=0.125} & {\text { (b) } 7^{3}=343}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Logarithmic Form

To express an equation in logarithmic form, we need to understand the inverse relationship between exponentiation and logarithms. For any expression of form \(a^b = c\), the logarithmic form is \(\log_a c = b\). Here, \(a\) is the base, \(c\) is the result, and \(b\) is the exponent.

Step 2: Converting Part (a) to Logarithmic Form

We have the equation \(4^{-3/2} = 0.125\). Using the formula \(\log_a c = b\), we identify \(a = 4\), \(b = -3/2\), and \(c = 0.125\). Therefore, the logarithmic form is \(\log_4 0.125 = -\frac{3}{2}\).

Step 3: Converting Part (b) to Logarithmic Form

We have the equation \(7^3 = 343\). Again, using \(\log_a c = b\), we identify \(a = 7\), \(b = 3\), and \(c = 343\). Therefore, the logarithmic form is \(\log_7 343 = 3\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation. It involves raising a number, called the base, to the power of an exponent. The result of this process is called the power. To represent it simply, if you have a base of 4 and an exponent of \(-3/2\), you write it as \(4^{-3/2}\). In this case, exponentiation helps us understand expressions involving repeated multiplication or division.

• The expression \(b^e\) means the base \(b\) is multiplied by itself \(e\) times.
• If the exponent is negative, like in \(4^{-3/2}\), it means the operation involves division.
• Fractional exponents such as \(-3/2\) represent roots, with the numerator indicating the power and the denominator the root to take.

Learning exponentiation is crucial for advancing in mathematics. It lays the foundation for understanding more complex operations and relationships in math, such as the inverse relationships involving logarithms.

Inverse Relationship

The inverse relationship is a concept where two operations do opposite things to each other. In mathematics, exponentiation and logarithms are inverse operations. This means that what one operation does, the other operation can undo.

• If you raise a number to a certain power in exponentiation, taking the logarithm with that same base will return the exponent.
• It is similar to how addition and subtraction are inverses, or multiplication and division are opposites.

For example, if \(7^3 = 343\), the logarithm will reverse the process to give us \(3\). This inverse relationship between exponentiation and logarithms helps to solve equations where the unknown is an exponent, making the concept of logarithms particularly useful.

Logarithms

Logarithms are a mathematical tool that allows us to find the exponent to which a base must be raised to produce a given number. The logarithmic form of an expression such as \(a^b = c\) is written as \(\log_a c = b\). Here, \(a\) is the base, \(c\) is the result called the antilogarithm, and \(b\) is the exponent or logarithm of \(c\) to the base \(a\).

• The logarithm asks "how many times do I have to multiply the base to get this number?"
• In the problem, converting \(4^{-3/2} = 0.125\) to logarithmic form gives \(\log_4 0.125 = -\frac{3}{2}\).
• Similarly, \(7^3 = 343\) becomes \(\log_7 343 = 3\), showing the power needed to reach the number 343 with base 7.

This ability to convert from and to logarithms is widely used in fields like engineering, science, finance, and more.

Base and Exponent

Understanding the base and exponent is crucial when working with exponential and logarithmic expressions. Every exponential expression consists of these two key components:

• Base (\(a\)): This is the number that is being repeatedly multiplied.
• Exponent (\(b\)): This indicates how many times the base is multiplied by itself. If it's a fraction or negative, more complex operations like roots and reciprocals are involved.

For instance, in \(4^{-3/2}\), \(4\) is the base, and \(-3/2\) is the exponent, telling us to take the reciprocal and a root of the base. In logarithmic form, the base stays the same, but we find the exponent by another method.

• In the equation \(7^3 = 343\), \(7\) is the base and \(3\) is the exponent.
• The logarithmic form reveals that multiplying the base \(7\) by itself \(3\) times equals \(343\).

Grasping how base and exponent interact deepens comprehension of both exponential and logarithmic functions.