/*! 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 15 For each statement, write an equ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

For each statement, write an equivalent statement in exponential form. $$\log _{10} 0.001=-3$$

Short Answer

Expert verified
The equivalent exponential form is \(10^{-3} = 0.001\).

Step by step solution

01

Understand Logarithmic Notation

The expression \(log_{10} 0.001 = -3\) means that 10 raised to what power gives 0.001. Here, the base of the logarithm is 10, and the argument is 0.001.
02

Apply the Definition of Logarithms

The logarithmic expression \(\log_{b}(a) = c\) can be rewritten in its equivalent exponential form as \(b^c = a\). Here, \(b = 10\), \(c = -3\), and \(a = 0.001\).
03

Rewrite in Exponential Form

Using the formula from Step 2, rewrite the logarithmic equation \(\log_{10} 0.001 = -3\) in exponential form as \(10^{-3} = 0.001\). This shows that raising 10 to the power of -3 results in 0.001.

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 Notation
Logarithmic notation is a way to express the relationship between exponents and their results. In the expression \(\log_{10} 0.001 = -3\), the notation itself communicates a lot. It means "the power to which 10 must be raised to result in 0.001 is -3." This is a compact way to express what would otherwise be an involved mathematical calculation. By understanding logarithmic notation, you're opening a door to simplifying the way we handle large or small numbers.
  • "log" represents the logarithm.
  • The subscript "10" is the base of the logarithm.
  • The main number, "0.001," is the argument or the result you're looking for.
  • The "-3" is the exponent, answering the question "Power to what?"
Logarithmic notation is like a language that makes these comparisons more efficient and understandable. It transforms multiplicative relationships into additive ones by focusing on the exponent.
Logarithm Base
The base in a logarithm is one of its most crucial parts and fundamentally influences the calculations. In our example \(\log_{10} 0.001 = -3\), the base is 10. The base tells us what number is being repeatedly multiplied by itself. In simpler terms, it answers what number needs to be raised to a particular power.

Common bases include
  • Base 10, which is often used in scientific calculations and is called the "common logarithm." It simplifies multiplying large numbers by using powers of ten.
  • Base \(e\), fundamental in calculus and known as the "natural logarithm."
  • Base 2, frequently used in computer science, representing binary code.
Understanding the base gives you a foundation for converting logarithmic notation into an exponential expression, as we shall see in the next section.
Exponential Expression
An exponential expression uses a base and an exponent to describe a number. When you see something like \(10^{-3} = 0.001\), it's showing how many times you multiply 10 by itself to reach 0.001. This is what the logarithmic notation was expressing in shorthand.

The conversion from logarithmic to exponential form is straightforward once you have identified:
  • The base \(b\), which is 10 in this case.
  • The exponent \(c\), which is the -3.
  • The result \(a\), which the base raised to the exponent should equal, here being 0.001.
Rewriting the expression from logarithmic to exponential helps visualize the operation as multiplication, offering a clearer perspective of how numbers relate and interact. This simplified view allows for better understanding and manipulation of equations.

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

Suppose that the cost of photovoltaic cells each year after 1980 was \(75 \%\) as much as the year prior. If the cost was \(\$ 30 /\) watt in \(1980,\) model their price in dollars with an exponential function, where \(x\) corresponds to years after \(1980 .\) Then estimate the year when the price of photovoltaic cells was \(\$ 1.00\) per watt.

Estimate the doubling time of an investment earning \(2.5 \%\) interest if interest is compounded (a) quarterly; (b) continuously.

Suppose that when a ball is dropped, the height of its first rebound is about \(80 \%\) of the initial height that it was dropped from, the second rebound is about \(80 \%\) as high as the first rebound, and so on. If this ball is dropped from 12 feet in the air, model the height in feet of each rebound with an exponential function \(H(x),\) where \(x=0\) represents the initial height, \(x=1\) represents the height on the first rebound, and so on. Find the height of the third rebound. Determine which rebound had a height of about 2.5 feet.

Suppose \(f(r)\) is the volume (in cubic inches) of a sphere of radius \(r\) inches. What does \(f^{-1}(5)\) represent?

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. \(\log _{10} x=x-2\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete 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.