/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Write each equation in exponenti... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 4

Write each equation in exponential form. \(\log _{3} 81=4\)

Short Answer

Expert verified
The exponential form is \( 3^4 = 81 \).

Step by step solution

01

Understand the Logarithmic Form

The given equation is \( \log _{3} 81 = 4 \). This represents a logarithmic equation where 3 is the base of the logarithm, 81 is the result, and 4 is the exponent to which the base must be raised to yield the result.
02

Convert to Exponential Form

To convert the logarithmic form to exponential form, recall that \( \log_b{a} = c \) can be rewritten as \( b^c = a \). In this case, \( \log_3{81} = 4 \) becomes \( 3^4 = 81 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Exponential Form
Converting a logarithmic equation into its exponential form is a straightforward yet crucial concept in understanding logarithmic functions.
The exponential form gives us a clear view of what the logarithmic equation is expressing.
In the exponential form, the equation is structured as \( b^c = a \):
  • \( b \) is the base.
  • \( c \) is the exponent.
  • \( a \) is the result.
For the equation \( \log_3{81} = 4 \), converting it to its exponential form, we express it as \( 3^4 = 81 \).
This form tells us that when 3 is raised to the power of 4, it equals 81.
Understanding this switch between forms enriches one's grasp of both the functions and how they relate to each other.
Logarithmic Equation
A logarithmic equation is a useful tool for solving problems involving exponential relationships.
The structure of a logarithmic equation is \( \log_b{a} = c \):
  • \( b \) is the base of the logarithm, which is the number being repeatedly multiplied.
  • \( a \) is the answer we would get from the exponential form.
  • \( c \) is the power or exponent needed for the base to reach the result \( a \).
In our example, \( \log_3{81} = 4 \), we see that the base 3 needs to be raised to the power of 4 to produce 81.
This highlights the inverse relationship logarithms have with exponentials, allowing us to solve for unknowns in various mathematical contexts.
Base of a Logarithm
The base of a logarithm is a fundamental part of understanding how logarithmic functions operate.
The base is the number that is being raised to a power to result in another number and acts as a 'pivot' in handling exponential transformations.
In the equation \( \log_b{a} = c \):
  • \( b \) is the base.
  • These functions tell us how many times we need to multiply the base by itself to achieve a certain number.]
For instance, in \( \log_3{81} = 4 \), the base is 3, indicating how many times 3 must be multiplied together to result in 81.
This concept is vital as it forms the foundation for transitioning between logarithmic and exponential equations effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use log \(_{3} 2 \approx 0.6309\) and \(\log _{3} 7 \approx 1.7712\) to approximate the value of each expression. \(\log _{3} 18\)

Use a calculator to evaluate each expression to four decimal places. \(e^{0.35}\)

An equation for loudness \(L\) in decibels, is \(L=10 \log _{10} R\) where \(R\) is the relative intensity of the sound. Solve \(130=10 \log _{10} R\) to find the relative intensity of a fireworks display with a loudness of 130 decibels.

Sketch the graph of each function. Then state the function's domain and range. $$ y=-\left(\frac{1}{5}\right)^{x} $$

For Exercises 15 and \(16,\) use the following information. Bacteria usually reproduce by a process known as binary fission. In this type of reproduction, one bacterium divides, forming two bacteria. Under ideal conditions, some bacteria reproduce every 20 minutes. Write the equation for modeling the exponential growth of this bacterium.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.