/*! 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 2 Write an equivalent exponential ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 2

Write an equivalent exponential equation. $$\log _{3} 81=4$$

Short Answer

Expert verified
The equivalent exponential equation is \( 3^{4} = 81 \).

Step by step solution

01

- Identify the logarithmic form

The given equation is in logarithmic form: \ \( \log_{3} 81 = 4 \). This means that the logarithm base is 3, the argument is 81, and the result is 4.
02

- Recall the definition of logarithms

The definition of a logarithm \( \log_{b} a = c \) states that \( b^{c} = a \). In this case, \( b = 3 \), \( a = 81 \), and \( c = 4 \).
03

- Rewrite the equation

Using the definition of logarithms, convert \( \log_{3} 81 = 4 \) into its exponential form: \ \( 3^{4} = 81 \).

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.

Logarithmic Equations
Logarithmic equations are mathematical expressions that involve the logarithm function.
The logarithm function helps us to determine the power to which a certain number, called the base, must be raised to produce another number.
In a logarithmic equation like \( \log_{3} 81 = 4 \), the base is 3, the argument is 81, and the result is 4.
This is read as 'the logarithm of 81 with base 3 equals 4'.
These equations are useful for solving problems where the value of the exponent is unknown.
When presented with a logarithmic equation, the goal is often to convert it to an exponential equation for easier manipulation and solution.
Exponential Equations
Exponential equations involve variables in the exponent.
These types of equations take the form \( b^c = a \), where \( b \) is the base, \( c \) is the exponent, and \( a \) is the result.
For example, \( 3^4 = 81 \) shows that raising 3 to the power of 4 results in 81.
When solving an exponential equation, the objective can be to isolate the exponent or determine the base that achieves the given result when raised to the exponent.
These equations are common in growth and decay problems, such as population growth or radioactive decay.
Logarithm Definition
Understanding logarithms is key to mastering both logarithmic and exponential equations.
The logarithm of a number is the exponent to which the base must be raised to produce that number.
This relationship is written as \( \log_{b} a = c \), which is read as 'the logarithm of a with base b equals c'.
Essentially, it tells us that \( b^c = a \).
Logarithms help in solving for exponents in equations where direct computation might be challenging.
They are also pivotal in various fields like engineering, science, and finance due to their ability to simplify complex calculations.
Conversion Between Forms
Converting between logarithmic and exponential forms is a fundamental skill in algebra.
The logarithmic form \( \log_{b} a = c \) can be converted to the exponential form \( b^c = a \) and vice versa.
This process is done using the definition of logarithms, which states that the logarithm of a number is the power to which the base must be raised to generate that number.
In the provided exercise, \( \log_{3} 81 = 4 \) is converted to its exponential form \( 3^4 = 81 \), demonstrating the equivalence between the two forms.
This conversion is helpful in solving equations, especially when the logarithmic form is challenging to manipulate directly.

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

The magnitude \(R\) (measured on the Richter scale) of an earthquake of intensity \(I\) is defined as \(R=\log \frac{I}{I_{0}},\) where \(I_{0}\) is the minimum intensity (used for comparison). (The exponential form of this definition is given in Exercise \(44 .)\) a) Find the rate of change \(d R / d I\). b) Interpret the meaning of \(d R / d I\)

To what exponential growth rate per hour does a growth rate of \(100 \%\) per day correspond?

Differentiate. $$f(x)=15^{x}$$

Differentiate. $$y=\ln \frac{x^{5}}{(8 x+5)^{2}}$$

Find the equation of the line tangent to the graph of \(y=\ln \left(4 x^{2}-7\right)\) at \(x=2\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics 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.