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Chapter 1: Problem 258

For the following exercises, write the equation in equivalent logarithmic form. $$ \left(\frac{1}{3}\right)^{3}=\frac{1}{27} $$

Short Answer

Expert verified
\( \log_{\frac{1}{3}} \frac{1}{27} = 3 \)

Step by step solution

01

Identify the Given Exponential Form

The equation \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \) is given in exponential form. Here, the base is \( \frac{1}{3} \), the exponent is \( 3 \), and the result is \( \frac{1}{27} \).
02

Structure of Logarithmic Form

Recall the structure for converting an exponential equation to logarithmic form: If \( a^b = c \), then in logarithmic form, it is \( \log_a c = b \).
03

Apply Structure to Given Equation

Apply the structure identified in Step 2 to the given equation \( \left( rac{1}{3}\right)^{3}=\frac{1}{27} \). Here, \( a = \frac{1}{3} \), \( b = 3 \), and \( c = \frac{1}{27} \).
04

Write the Equivalent Logarithmic Form

Convert the exponential equation to logarithmic form using the formula from Step 2: \[ \log_{\frac{1}{3}} \frac{1}{27} = 3 \]
05

Verify the Conversion

Verify that our conversion is accurate. The expression \( \log_{\frac{1}{3}} \frac{1}{27} = 3 \) indicates that raising the base \( \frac{1}{3} \) to the power of 3 results in \( \frac{1}{27} \), which matches our original equation.

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Key Concepts

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

Exponential Equation
An exponential equation is a mathematical expression in which a constant base is raised to a variable exponent. This type of equation is fundamental because it describes rapid growth or decay, and it's prevalent in various mathematical and real-world situations.
The given example, \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \), is an exponential equation. Here, the base is \( \frac{1}{3} \), and it is raised to the power of 3, which is the exponent. The result obtained is \( \frac{1}{27} \).
Exponential equations can be represented in the general form \( a^b = c \), where:
  • \( a \) is the base,
  • \( b \) is the exponent,
  • \( c \) is the result.
Understanding this form is crucial when dealing with any problem involving exponential growth, decay, or transformation into logarithmic form.
Base-Exponent-Result Relationship
The relationship between the base, exponent, and result is a cornerstone of exponential equations. This connection is also vital when converting exponential equations into logarithmic form.
Let's break it down with the equation \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \):
  • Base (\( \frac{1}{3} \)): The number that is repeatedly multiplied by itself.
  • Exponent (3): Indicates how many times the base is multiplied.
  • Result (\( \frac{1}{27} \)): The final product obtained after performing the exponential operation.
When these components interact, they create an equation that can be translated into logarithmic form, offering a different perspective on the same mathematical relationship.
Converting Logarithmic Expressions
Conversion from exponential to logarithmic form is a key skill in mathematics, making complex equations easier to handle. The process involves understanding the structure that connects these two concepts.
Any exponential equation \( a^b = c \) can be rewritten in logarithmic form as \( \log_a c = b \). This transformation highlights the exponent as a result of the logarithm of the base and the result.
For the equation \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \), you can convert it to \( \log_{\frac{1}{3}} \frac{1}{27} = 3 \). This states that 3 is the power to which \( \frac{1}{3} \) must be raised to get \( \frac{1}{27} \).
Understanding how to switch between these forms is essential for solving equations and grasping more advanced mathematical concepts with ease.

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Most popular questions from this chapter

For the following exercises, write the equation in equivalent logarithmic form. $$ \sqrt[3]{64}=4 $$

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season. In reality, the overall population is most likely increasing or decreasing throughout each year. Let's reformulate the model as \(P(t)=82.5-67.5 \cos [(\pi / 6) t]+t, \quad\) where \(t\) is time in months \((t=0 \text { represents January } 1)\) and \(P\) is population (in thousands). When is the first time the population reaches \(200,000 ?\)

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. $$y=3 \cos (2 x+3)$$

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. $$y=-3 \sin (\pi x+2)$$

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=\log x-1 $$
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