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Chapter 5: Problem 37

Convert to a logarithmic equation. \(8^{1 / 3}=2\)

Short Answer

Expert verified
log_{8}(2) = 1/3

Step by step solution

01

- Understand the given equation

The equation provided is an exponential form: 8^{1/3}=2.
02

- Recall the relationship between exponential and logarithmic forms

The general form of an exponential equation is a^{b} = c. Its equivalent logarithmic form is log_{a}(c) = b.
03

- Identify the components

In our equation, a = 8, b = 1/3, and c = 2.
04

- Formulate the logarithmic equation

Using the relationship from Step 2: log_{8}(2)=1/3. This is the logarithmic form of the given exponential equation.

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Key Concepts

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

Exponential Functions
Exponential functions are a kind of mathematical function denoted as in the form of \(a^b\). Here, \(a\) is the base and \(b\) is the exponent. In the given problem, \(8^{1/3}=2\), the base \(a\) is 8, and the exponent \(b\) is \(1/3\). The result or the power is 2.
Exponential functions are essential in many areas like growth processes, compound interest calculations, and more.
It's important to note:
  • The base must always be a positive real number.
  • The exponent can also be a fraction, meaning the root (such as \(1/3\) representing the cube root).
  • For exponential growth or decay equations, the form may look like \(y = a \cdot b^x\).
Logarithmic Functions
Logarithmic functions serve as the inverse of exponential functions. They answer the question: To what power should we raise the base \(a\) to get the result \(c\)? Mathematically, the logarithmic form of an exponential equation \(a^b = c\) is written as \(log_a(c) = b\).
The specific logarithmic function in the given example is \(log_8(2) = 1/3\), which tells us that raising 8 to the power of \(1/3\) results in 2.
Key points to remember:
  • The base \(a\) of a logarithm must be positive and not equal to 1.
  • \

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Most popular questions from this chapter

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