Problem 16 - A Graphical Approach to College Algebra

Question

For each statement, write an equivalent statement in exponential form. $$\log _{3} \sqrt[3]{9}=\frac{2}{3}$$

Step-by-Step Solution

Answer

The equivalent exponential form is \( 3^{\frac{2}{3}} = \sqrt[3]{9} \).

Step 1: Understanding the Logarithm

We have the logarithmic statement \( \log_{3} \sqrt[3]{9} = \frac{2}{3} \), which says the power to which the base \(3\) must be raised to get \(\sqrt[3]{9}\) is \(\frac{2}{3}\).

Step 2: Converting to Exponential Form

In exponential form, the logarithmic equation \( \log_{3} \sqrt[3]{9} = \frac{2}{3} \) becomes an equation of the form: \[ 3^{\frac{2}{3}} = \sqrt[3]{9} \].

Step 3: Verifying the Conversion

Let's verify by evaluating both sides: - For the left side, \(3^{\frac{2}{3}}\) can be rewritten using root and power laws as \(\left(3^2\right)^{\frac{1}{3}} = 9^{\frac{1}{3}} = \sqrt[3]{9}\).- For the right side, it directly represents \(\sqrt[3]{9}\).Both sides match, confirming the equivalency in exponential form.

Key Concepts

Logarithmic Equations

Logarithmic equations involve logarithms, which are the inverse operations of exponents. A logarithm answers the question: "To what power must we raise the base to get a certain number?" In the equation \( \log_{3} \sqrt[3]{9} = \frac{2}{3} \), the base is 3. We're finding the power (or exponent) needed to convert 3 into \( \sqrt[3]{9} \).
Understanding logarithmic equations requires recognizing that they are another way to express exponential relationships. You can think of them as asking, "How many times do we multiply the base by itself to achieve the result?"

• The base, like 3 in our equation, is what you repeatedly multiply.
• The result (or argument) is \( \sqrt[3]{9} \).
• The answer is the exponent, which in this case is \( \frac{2}{3} \).

These equations are crucial for solving problems involving exponential growth, compound interest, and scientific data analysis.

Exponents

Exponents involve raising a number, called the base, to the power of another number. This indicates how many times you multiply the base by itself. For instance, in the term \( 3^{\frac{2}{3}} \), 3 is the base and \( \frac{2}{3} \) is the exponent.
Conversions between logarithmic and exponential forms require understanding of how these two forms are interconnected. In our exercise, the conversion translates the log statement \( \log_{3} \sqrt[3]{9} = \frac{2}{3} \) into the exponential form \( 3^{\frac{2}{3}} = \sqrt[3]{9} \).

• When an exponent is a fraction, like \( \frac{2}{3} \), it can represent both roots and powers.
• The numerator of the fraction (2) represents the power.
• The denominator (3) represents the root.

Understanding exponents is essential for tackling more complex problems in algebra and calculus, as they underpin many mathematical transformations.

Radicals

Radicals involve roots of numbers and are a key part of simplifying expressions. A radical indicates you are finding the root of a number. In the problem, \( \sqrt[3]{9} \) means finding the cube root of 9.
Radicals often appear with fractional exponents, bridging a crucial link between radical and exponential expressions. For instance, the equation we examined shows that \( \sqrt[3]{9} \) can be rewritten as \( 9^{\frac{1}{3}} \), showcasing this relationship.

• Radicals are written using the radical sign \( \sqrt{} \), while the index of the radical is shown in the upper left.
• In expressions, the index indicates which root you're dealing with, like square root (2) or cube root (3).
• Radical expressions can often be converted to expressions with fractional exponents, providing flexibility in mathematical operations.

Mastering radicals strengthens your ability to handle algebraic manipulations and complex numbers.