Problem 9 - Algebra and Trigonometry

Question

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{8} 2=\frac{1}{3}} & {\text { (b) } \log _{2}\left(\frac{1}{4}\right)=-3}\end{array} $$

Step-by-Step Solution

Answer

(a) \( 8^{\frac{1}{3}} = 2 \); (b) \( 2^{-3} = \frac{1}{4} \).

Step 1: Understand the Logarithmic Equation (a)

For part (a), we have the logarithmic equation \( \log_8 2 = \frac{1}{3} \). We want to express this in exponential form. The logarithmic statement \( \log_b a = c \) is equivalent to the exponential statement \( b^c = a \).

Step 2: Convert Equation (a) to Exponential Form

Using the equivalence between logarithmic and exponential forms, \( \log_8 2 = \frac{1}{3} \) can be rewritten as \( 8^{\frac{1}{3}} = 2 \).

Step 3: Understand the Logarithmic Equation (b)

For part (b), we have \( \log_2 \left(\frac{1}{4}\right) = -3 \). Again, we want to convert this into exponential form using the same principle as before.

Step 4: Convert Equation (b) to Exponential Form

The equation \( \log_2 \left(\frac{1}{4}\right) = -3 \) translates to \( 2^{-3} = \frac{1}{4} \) in exponential form, since the base raised to the power of the logarithm equals the argument.

Key Concepts

Logarithmic Equation

A logarithmic equation is a way to express the power to which a number, called the base, must be raised to yield a given number. It is often written in the form \( \log_b a = c \), where \( b \) is the base, \( a \) is the result after raising \( b \) to a power, and \( c \) is the exponent. Logarithmic equations allow us to solve for the exponent when the base and the result are known.

Let's consider two examples:

• Example 1: \( \log_8 2 = \frac{1}{3} \)
• Example 2: \( \log_2 \left(\frac{1}{4}\right) = -3 \)

In these examples, the logarithmic equations provide a way to determine the powers \( \frac{1}{3} \) and \(-3\) necessary to transform the bases \(8\) and \(2\) into the results \( 2 \) and \( \frac{1}{4} \), respectively. Understanding this construct is essential for anyone looking to manipulate and convert between logarithmic and exponential forms effectively.

Base and Exponent

In mathematics, the terms "base" and "exponent" play crucial roles in expressions involving powers and logarithms. The base is the number that is multiplied by itself a certain number of times as dictated by the exponent. In other words, the exponent tells us how many times the base is used as a factor.

For example, in the equation \( 8^{\frac{1}{3}} \), the number \( 8 \) is the base and \( \frac{1}{3} \) is the exponent. This specific scenario implies the cube root of \( 8 \), resulting in a value of \( 2 \), which correlates with the logarithmic equation \( \log_8 2 = \frac{1}{3} \).

Similarly, in \( 2^{-3} = \frac{1}{4} \), the base is \( 2 \), and \(-3\) is the exponent. Here, the negative exponent indicates the reciprocal, corresponding to raising \( 2 \) to the power of \( 3 \) and then taking the reciprocal. Hence, \( 2^{-3} \) results in \( \frac{1}{4} \). Understanding bases and exponents is essential for converting logarithmic equations into exponential forms.

Equivalence of Logarithmic and Exponential Forms

The equivalence between logarithmic and exponential forms is a vital skill in mathematics. This concept simplifies the transition from logarithmic equations to exponential equations and vice versa. A standard template for this conversion is: if \( \log_b a = c \), then it is equivalent to the exponential form \( b^c = a \).

Let's put this into practice with our earlier examples:

• For \( \log_8 2 = \frac{1}{3} \), the equivalent exponential form is \( 8^{\frac{1}{3}} = 2 \).
• For \( \log_2 \left(\frac{1}{4}\right) = -3 \), the equivalent exponential form is \( 2^{-3} = \frac{1}{4} \).

Recognizing this equivalence helps in solving equations involving powers. It also makes understanding complex mathematical relationships easier, as you can switch between forms to suit the problem context. With practice, converting between these forms becomes an intuitive and powerful mathematical tool.