/*! 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 8 Write an equivalent exponential ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

Write an equivalent exponential equation. $$-\log _{b} V=w$$

Short Answer

Expert verified
\(b^{-w} = V\)

Step by step solution

01

Understand Logarithmic and Exponential Forms

Recall that \(\text{if} \log_b(x) = y\), then \(b^y = x\). The logarithmic form and the exponential form are inverses of each other.
02

Rewrite the Given Logarithmic Equation

The given equation is \(-\log_b V = w\). To make it easier to convert to exponential form, isolate the logarithm: \minus(-\log_b V = w)\ implies \(\log_b V = -w\).
03

Convert to Exponential Form

Using the rule that \(\log_b(x) = y\) means \(b^y = x\), apply this to the rearranged equation \(\log_b V = -w\). This gives \(b^{-w} = V\).

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 Form
Logarithmic form helps us understand the relationship between numbers and their powers. It's written as \(\text{\log}_b(x) = y\). In plain language, this means 'What exponent do we need on base b to get x.' Imagine it like finding the missing piece in a multiplication puzzle. Here, \(b\) is the base, \(x\) is the result, and \(y\) is the exponent.

For example, if \(\text{\log}_2(8) = 3\), it means that you need to raise 2 to the power of 3 to get 8.

The logarithmic form is handy because it allows us to work backwards from the result to find the original base and exponent. It鈥檚 the opposite of raising a number to a power.
Exponential Form
Exponential form is another way to describe the relationship between numbers. It鈥檚 written as \(b^y = x\). This states that if you take a base \(b\) and raise it to a power \(y\), you鈥檒l get \(x\).

For example, \(2^3 = 8\) means that 2 raised to the power of 3 equals 8.

Converting between exponential and logarithmic forms hinges on recognizing that they are just different perspectives of the same relationship. If you know \(b^y = x\), you can always reframe it as \(\text{\log}_b(x) = y\). This makes solving different kinds of problems much easier since you can utilize the format that suits the problem best.
Logarithmic Equations
Logarithmic equations include a logarithm of a variable. They follow the general form \(\text{\log}_b(x) = y\). To solve for the variable, you often convert the log form to exponential form. This helps to simplify and solve the equation.

Here鈥檚 the exercise example explained:
  • Start with \(-\text{\log}_b(V) = w\).
  • Isolate the logarithm: \(\text{\log}_b(V) = -w\).
  • Apply the conversion rule: \(\text{\log}_b(V) = -w\) means \(b^{-w} = V\).

This process shows how to switch from logarithmic to exponential form and solve for the original variable easily. By understanding these principles, you can tackle a variety of logarithmic and exponential problems more confidently.

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 Hullian learning model asserts that the probability p of mastering a task after \(t\) learning trials is approximated by \(p(t)=1-e^{-k t}\) where \(k\) is a constant that depends on the task to be learned. Suppose that a new dance is taught to an aerobics class. For this particular dance, the constant \(k=0.28\) a) What is the probability of mastering the dance's steps in 1 trial? 2 trials? 5 trials? 11 trials? 16 trials? 20 trials? b) Find the rate of change, \(p^{\prime}(t)\) c) Sketch a graph of the function.

The intensity of an earthquake is given by \(I=I_{0} 10^{R},\) where \(R\) is the magnitude on the Richter scale and \(I_{0}\) is the minimum intensity, at which \(R=0,\) used for comparison. a) Find \(I,\) in terms of \(I_{0},\) for an earthquake of magnitude 7 on the Richter scale. b) Find \(I\), in terms of \(I_{0},\) for an earthquake of magnitude 8 on the Richter scale. c) Compare your answers to parts (a) and (b). d) Find the rate of change dI/dR. e) Interpret the meaning of \(d I / d R\)

For the demand function given in each,Find the following. a) The elasticity b) The elasticity at the given price, stating whether the demand is elastic or inelastic c) The value(s) of \(x\) for which total revenue is \(a\) maximum (assume that \(x\) is in dollars) $$q=D(x)=500-x ; \quad x=38$$

Differentiate. $$f(x)=e^{x / 2} \cdot \sqrt{x-1}$$

Explain how the Rule of 70 could be useful to someone studying inflation.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.