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Chapter 9: Problem 28

Write each as a logarithmic equation. See Example 2. $$ 4^{1 / 3}=\sqrt[3]{4} $$

Short Answer

Expert verified
\(\log_4 (\sqrt[3]{4}) = \frac{1}{3}\)

Step by step solution

01

Identify the Base and Result

In the expression \(4^{1/3} = \sqrt[3]{4}\), identify the base and result. In this case, the base is 4, and the result is \(\sqrt[3]{4}\).
02

Determine the Exponent

Notice that the form of the expression is \(a^b = c\). Therefore, the exponent here is \(\frac{1}{3}\).
03

Convert to Logarithmic Form

The equation \(a^b = c\) can be written in logarithmic form as \(\log_a c = b\). Substitute the base (4), result (\(\sqrt[3]{4}\)), and the exponent (\(\frac{1}{3}\)) to get \(\log_4 (\sqrt[3]{4}) = \frac{1}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Base and Exponent
The concepts of base and exponent are fundamental in mathematics. They are crucial for understanding equations involving powers. In an expression like \(4^{1/3}\), the number 4 is known as the "base" and the fraction \(1/3\) is the "exponent". The base is the number that is being multiplied by itself, whereas the exponent indicates how many times the base is multiplied. In our case:
  • The base is 4, signifying that this is the number we start with.
  • The exponent \(1/3\) signifies that we are taking the cube root of the base rather than multiplying it a straightforward number of times.
Essentially, \(4^{1/3}\) asks: "What number, multiplied by itself three times, equals 4?" By understanding these components, we can begin to unravel more complex equations involving powers.
Converting Expressions Between Forms
Converting expressions from one form to another is a valuable skill in algebra and calculus. You will often need to shift between exponential and root forms to simplify problems or solve equations. For instance, the expression \(4^{1/3}\) appears mysterious at first glance. However, it can be converted into a more familiar form: the cube root of 4, or \(\sqrt[3]{4}\).
  • In exponential form, the base is raised to a fraction \( \frac{1}{3} \).
  • In root form, this is the same as asking for the cube root.
This conversion allows you to understand that taking the cube root is essentially finding the number which, when used as a factor three times (multiplied by itself twice more), results in the base. Understanding these conversions is key when manipulating and solving logarithmic equations.
Applying Logarithmic Form
Logarithms are a unique way of expressing exponentiation, pivotal in solving equations where identifying powers isn't straightforward. To switch an exponential equation to a logarithmic one, you employ the formula \(a^b = c\) which transforms to \(\log_a c = b\). So, when you see \(4^{1/3} = \sqrt[3]{4}\), turning this into the logarithmic form involves:
  • Using the base, "4", from the original equation.
  • The value \(\sqrt[3]{4}\) becomes what you're taking the logarithm of.
  • The answer, or "\(b\)", comes out as \(1/3\).
Thus, you arrive at the logarithmic equation \(\log_4 (\sqrt[3]{4}) = \frac{1}{3}\). This form describes the operation more explicitly, showing exactly how exponentiation and logarithms are inherently related. By learning to convert between these expressions, you enhance your ability to tackle mathematical challenges efficiently.

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Most popular questions from this chapter

Solve each equation for \(y\). $$x=-6 y$$

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