Problem 54 - Precalculus : Building Concepts and Connections

Question

Use the definition of a logarithm to solve for \(x\). $$\log _{6} x=-2$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \( \log _{6} x = -2 \) is \( x = 1/36 \)

Step 1: Understanding the Problem

First, make sure you understand that the problem asks to solve the logarithmic equation \( \log _{6} x = -2 \) for \( x \). It's important to grasp the essence of logarithmic functions. In this equation, \( 6 \) is the base, \( -2 \) is the exponent and \( x \) is the result.

Step 2: Conversion to Exponential Form

Convert the logarithmic equation to an exponential equation. That means, if \( \log _{b} a = c \), then this is the same as \( b^c = a \). Applying this rule here, \( 6^{-2} = x \).

Step 3: Simplifying the Exponential Equation

Next, solve the equation \( 6^{-2} = x \) for \( x \). Remember that any number to the power of \(-n\) is equivalent to \( 1 \) divided by the number to the power of \( n \). Thus, \( 6^{-2} = 1/6^2 \).

Step 4: Calculate the Value of \( x \)

Calculate the value of \( x \) as \( 1/6^2 = 1/36 \). Thereby \( x = 1/36 \)

Key Concepts

Logarithm Definition

A logarithm is essentially an operation that determines how many times we must use one number to multiply itself to get another number. In simpler terms, if we have the logarithm \(\log_{b}a = c\), it tells us that the base \(b\) must be raised to the power \(c\) to get \(a\). So, logarithms are the inverse operations of exponentiation.

Let's apply this to the given problem. The equation \(\log _{6} x=-2\) is asking: to which power must we raise the number 6 to get the number \(x\)? Well, as per the definition of logarithm, it's assumed that \(6\) raised to the power of \( -2 \) will equal \(x\). This is a different way to view exponential relationships, focusing on the exponent rather than the result of the calculation. In our daily lives, logarithms are used in various fields such as science, engineering, and finance, to solve problems involving exponential growth or decay.

Exponential Form

The exponential form is a way of expressing a number using exponents or powers. It is the process of taking a base number and raising it to a certain power. The exponential form is rewritten from its corresponding logarithmic form, which we use to solve logarithmic equations. To clarify, if you have an equation in the form \(\log _{b} a = c\), it can be converted into the exponential form as \(b^{c} = a\).

In our exercise, we've transformed the logarithmic equation \(\log _{6} x=-2\) to its exponential counterpart \(6^{-2} = x\). This conversion is crucial for solving logarithmic equations. After converting, it becomes a matter of simple exponentiation to find the value of \(x\). Understanding how to switch between logarithmic and exponential forms is not just useful for solving textbook problems; it's also a skill that applies to understanding exponential growth rates, such as those observed in populations, investments, and even in calculating compound interest.

Exponents and Powers

Exponents and powers are used to describe how many times a number, known as the 'base', is multiplied by itself. The exponent, often written as a superscript to the right of the base, signifies this repetition. Positive exponents like \(2^{3} = 2 \times 2 \times 2 = 8\) indicate direct multiplication. Negative exponents, like \(6^{-2}\) found in the exercise, indicate division—as in \(1/6^{2}\).

The notion of a negative exponent can be understood with the rule that anything to the power of \( -n \) is equal to \(1\) over that number to the power of \( n \) — in other words, \(a^{-n} = 1/a^{n}\). It's a powerful concept that simplifies multiplication and division involving large numbers. Mastery of exponents and powers is essential for higher-level mathematics, physics, and computer science. For instance, exponential notation makes dealing with large numbers, such as Avogadro's number in chemistry or light-years in astronomy, practical and manageable.