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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 \(b^{-w} = V\).

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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.

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Most popular questions from this chapter

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) $$q=D(x)=500-x ; \quad x=38$$

Iodine-131 has a decay rate of \(9.6 \%\) per day. The rate of change of an amount \(N\) of iodine- 131 is given by \(\frac{d N}{d t}=-0.096 N\) where \(t\) is the number of days since the decay began. a) Let \(N_{0}\) represent the amount of iodine-131 present at \(t=0 .\) Find the exponential function that models the situation. b) Suppose that \(500 \mathrm{g}\) of iodine- 131 is present at \(t=0\) How much will remain after 4 days? c) After how many days will half of the 500 g of iodine-131 remain?

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Differentiate. $$F(x)=-\frac{2}{3} e^{x^{2}}$$

The population of the United States in 1776 was about 2,508,000 In the country's bicentennial year, the population was about 216,000,000 a) Assuming an exponential model, what was the growth rate of the United States through its bicentennial year? b) Is exponential growth a reasonable assumption? Explain.
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