Problem 5 - Algebra and Trigonometry

Question

\(3-8\) Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{8} 2=\frac{1}{3}} & {\text { (b) } \log _{2}\left(\frac{1}{8}\right)=-3}\end{array} $$

Step-by-Step Solution

Answer

(a) \( 8^{\frac{1}{3}} = 2 \); (b) \( 2^{-3} = \frac{1}{8} \).

Step 1: Identify the components of the logarithmic equation

For part (a), the equation is \(\log_{8} 2 = \frac{1}{3}\). Here, 8 is the base, 2 is the argument, and \( \frac{1}{3} \) is the exponent of the equation. For part (b), the equation is \(\log_{2}\left(\frac{1}{8}\right) = -3\). Here, 2 is the base, \( \frac{1}{8} \) is the argument, and \(-3\) is the exponent.

Step 2: Rewrite as an exponential equation

To express the logarithmic equation in exponential form, use the definition: if \( \log_b a = c \), then \( b^c = a \). For part (a), this results in \( 8^{\frac{1}{3}} = 2 \). For part (b), this translates to \( 2^{-3} = \frac{1}{8} \).

Step 3: Simplify the exponential form

In part (a), cube root of 8 results in 2, confirming \( 8^{\frac{1}{3}} = 2 \). In part (b), \( 2^{-3} \) equals \( \frac{1}{8} \), confirming \( 2^{-3} = \frac{1}{8} \).

Key Concepts

Logarithmic Equations

Logarithmic equations involve variables that are part of a logarithm. Understanding the components of a logarithmic equation can be essential. A typical logarithmic equation is in the form of \( \log_b a = c \). Here:

• \( b \) is the base of the logarithm.
• \( a \) is the argument.
• \( c \) is the result, which represents the exponent that the base must be raised to, in order to yield the argument.

This structure helps to convert the logarithmic equation into its equivalent exponential form. Knowing how to identify these components is the first step to manipulating and understanding logarithmic equations. Let's look at the example \( \log_8 2 = \frac{1}{3} \). Here, the number 8 is the base, 2 is the argument, and \( \frac{1}{3} \) is the exponent. In practice, this tells us that raising 8 to the power of \( \frac{1}{3} \) yields 2.

Exponential Equations

An exponential equation is one where the variables appear as exponents. It's essentially the reverse form of a logarithmic equation. The general structure is \( b^c = a \), where:

• \( b \) is the base.
• \( c \) is the exponent.
• \( a \) is the result (or argument in logarithmic form).

For instance, converting the logarithmic equation \( \log_8 2 = \frac{1}{3} \) into an exponential equation, we get \( 8^{\frac{1}{3}} = 2 \). This implies that if you take the cube root of 8, you'll get 2. Similarly, for \( \log_2 \left(\frac{1}{8}\right) = -3 \), the converted exponential equation becomes \( 2^{-3} = \frac{1}{8} \). In this equation, we raise 2 to the power of -3, resulting in the fraction \( \frac{1}{8} \). Understanding these relationships allows you to solve and graph such equations efficiently.

Logarithms and Exponents

Logarithms and exponents are closely related mathematical concepts. Simply put, logarithms are the inverse operations of exponents. They allow you to determine the power or exponent needed on a base number to achieve a given number. This is why an equation like \( \log_b a = c \) is equivalent to \( b^c = a \).

• Exponents describe how many times a number, the base, is multiplied by itself.
• Logarithms tell you what exponent gives you a specific number from a base.

Understanding the relationship between logarithms and exponents can be a fundamental skill in mathematics and is widely applicable in fields like computer science, physics, and statistics. The base and the result-interchanging nature of these concepts make it easy to switch between their forms. With mastery of this concept, unraveling complex exponential and logarithmic expressions becomes a straightforward task.