Problem 14 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in exponential form. Do not use a calculator. $$\log _{3} \sqrt[3]{9}=\frac{2}{3}$$

Step-by-Step Solution

Answer

\( 3^{\frac{2}{3}} = \sqrt[3]{9} \)

Step 1: Identify the Logarithmic Equation

The given equation is \( \log_{3} \sqrt[3]{9} = \frac{2}{3} \). This equation is expressed in logarithmic form and represents a relationship between base 3, the argument \( \sqrt[3]{9} \), and the result \( \frac{2}{3} \).

Step 2: Convert the Logarithmic Form to Exponential Form

The general formula to convert a logarithmic equation \( \log_b(a) = c \) to exponential form is \( b^c = a \). Applying this to our given equation, we replace \( b \) with 3, \( a \) with \( \sqrt[3]{9} \), and \( c \) with \( \frac{2}{3} \).

Step 3: Write the Exponential Form

Utilizing the conversion, the exponential form of the equation is \( 3^{\frac{2}{3}} = \sqrt[3]{9} \). This illustrates that raising the base 3 to the power of \( \frac{2}{3} \) yields the cube root of 9.

Key Concepts

Logarithmic Form

Logarithmic form is a mathematical notation that is essentially the reverse of exponential notation. When you see an equation like \( \log_b(a) = c \), this means that \( b^c = a \). The base \( b \) is raised to the power of \( c \) to result in \( a \). This form is incredibly useful for solving equations where the unknown is an exponent. By transforming such problems into logarithmic form, they become more manageable.

A real-world analogy could be thinking of logarithms as undoing exponentiation, much like division undoes multiplication. When you encounter logarithms:

• The base \( b \) represents the number being raised to a power.
• The argument or result \( a \) is the number you end up with after raising the base \( b \) to the power \( c \).
• The exponent \( c \) shows how many times we use the base in multiplication to reach the argument \( a \).

Using our original problem, you can understand that each part of the logarithmic form has a unique and critical role.

Base and Exponent

The base and exponent are fundamental concepts in both exponential and logarithmic expressions. The base, often seen in expressions like \( b^c \), is the number that is multipled by itself a certain number of times. The exponent \( c \) determines how many times we multiply the base by itself.

In exponential form, \( b^c = a \):

• \( b \) is the base
• \( c \) is the exponent
• \( a \) is the result of the base raised to the exponent

Consider the example from the given exercise: \( 3^{\frac{2}{3}} = \sqrt[3]{9} \). In this case:

• The base is 3. This is the number being repeatedly multiplied.
• The exponent is \( \frac{2}{3} \). This specific fraction signifies not just multiplication but also roots, indicating a more complex operation.
• The result is \( \sqrt[3]{9} \), which is equivalent to 3 being raised to the power of \( \frac{2}{3} \).

Understanding these roles makes it easier to work with exponential equations and aids in converting them to logarithmic form and vice versa.

Cube Root

The cube root of a number \( a \) is a special type of root that represents a number which, when multiplied by itself three times, equals \( a \). It is written as \( \sqrt[3]{a} \). For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).

Cube roots are particularly interesting because they introduce a third dimension to the multiplication process, not unlike wrapping an object to fit into a cube. This operation is important in various fields, including geometry and physics, where volumetric calculations are frequent.

In our exercise, \( \sqrt[3]{9} \) represents a cube root. The expression \( 3^{\frac{2}{3}} = \sqrt[3]{9} \) implies converting a power notation into a root. The fractional exponent \( \frac{2}{3} \) signifies that we first find the cube root of 9, then raise it to the power of 2. It is a flexible and useful way to express numbers undergoing these calculations.