Question

Rewrite the equation in exponential form. (See Example 1.) \(\log _3 \frac{1}{3}=-1\)

Short answer

The given logarithm rewritten in an exponential form is \( 3^{-1} = \frac{1}{3} \)

Step-by-step solution

Understand the Problem

The given logarithmic equation is \( \log _3 \frac{1}{3}=-1 \). The base of the logarithm is 3, the argument of the logarithm is \( \frac{1}{3} \), and the result is -1.

Apply the Logarithm to Exponential Rule

The rule is in the form \( b^c = a \) for a logarithm in the form \( \log_b a = c \). Applying this rule to the given logarithm, where 3 is the base (b), \( \frac{1}{3} \) is the argument of the logarithm (a), and -1 is the result (c), we get \( 3^{-1} = \frac{1}{3} \)

Check Result

Verify that the result is correct by confirming that \( 3^{-1} = \frac{1}{3} \)

Key concepts

Logarithmic Equations

Understanding logarithmic equations is essential for many mathematical applications. In simple terms, a logarithm answers the question: 'To what exponent must we raise a base number to obtain another number?'. For instance, when we look at the equation \[\begin{equation} \log_3 \frac{1}{3} = -1 \end{equation}\], we interpret this as 'What power should 3 be raised to, to get \frac{1}{3}?'. The logarithmic equation provided in the example can be bewildering at first, but with the correct approach, one can see it simply represents an alternative way to express exponentiation.

Rewriting logarithmic equations into their exponential form allows us to see the relationship between the components more clearly. The step by step conversion offered is helpful, but to further improve understanding, students should practice converting several examples from logarithmic to exponential form until the process becomes intuitive.

Exponents

Exponents, also called powers, are a shorthand way of expressing repeated multiplication of a number by itself. In the equation \[\begin{equation} 3^{-1} = \frac{1}{3} \end{equation}\], the '-1' is the exponent, indicating that the inverse of 3 (which is \frac{1}{3}) is being considered. Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. Therefore, \[\begin{equation} 3^{-1} \end{equation}\], essentially instructs us to take the reciprocal of 3 and raise it to the power of 1. This concept is vital for understanding more complex mathematical ideas, and mastering the basics of exponents will pave the way for learning more intricate exponential functions.

Grasping the concept of exponents involves comprehending both positive and negative powers, as well as zero as an exponent. It's also beneficial to recognize patterns, such as any number to the power of zero is one, to solidify the foundation of exponent knowledge.

Logarithm Rules

The realm of logarithms is governed by a set of rules that make manipulating and solving logarithmic expressions much simpler. These rules include the product rule, quotient rule, power rule, and change of base formula. Understanding how to apply these rules can turn a seemingly complicated logarithmic equation into a more manageable form.

In the given exercise, the rule we focus on is the one that connects logarithms with exponents, allowing us to switch from logarithmic to exponential form. The rule states that \[\begin{equation} \log_b a = c \end{equation}\], can be written as \[\begin{equation} b^c = a \end{equation}\], where \(b\) is the base of the logarithm, \(a\) is the result, and \(c\) is the exponent to which the base is raised. By practicing this rule, along with the others mentioned, tackling logarithmic equations becomes less daunting, opening up a clearer path to solving them effectively.