/*! 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 136 What is a base-10 logarithm?... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 136

What is a base-10 logarithm?

Short Answer

Expert verified
A base-10 logarithm, also known as the common logarithm, is the inverse operation to exponentiation with base 10. It helps us find the exponent to which 10 has to be raised to obtain a given value. It is represented as \(\log_{10} b\) or simply as \(\log b\). For example, \(\log_{10} 1000 = 3\) or \(\log 1000 = 3\) since \(10^3 = 1000\).

Step by step solution

01

Definition of Logarithm

A logarithm is the inverse operation to exponentiation. It helps us find the exponent to which a specific base has to be raised to obtain a given value. In general form, if \(a^x = b\), then the logarithm of b with base a is \(x\), denoted as \(\log_a b = x\).
02

Base-10 Logarithm (Common Logarithm)

A base-10 logarithm, also known as the common logarithm, uses 10 as its base. It means that we are looking for the exponent to which 10 has to be raised to get the given number. The base-10 logarithm is represented as \(\log_{10} b\) or simply as \(\log b\).
03

Practical Example

Let's consider the number 1000. To find the base-10 logarithm of 1000, we need to figure out the exponent to which 10 needs to be raised to result in 1000. Since \(10^3 = 1000\), the base-10 logarithm of 1000 is 3, represented as \(\log_{10} 1000 = 3\) or \(\log 1000 = 3\).

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.

Logarithm Definition
Understanding the concept of logarithms is fundamental in various fields of mathematics and science. A logarithm answers the question: To what exponent must we raise a certain base number to get another number? Essentially, it's a way to solve for an unknown exponent in exponential equations.

For instance, consider the equation where a raised to the power of x equals b, written as \(a^x = b\). The logarithm of b with respect to base a, symbolized as \(\log_a b\), is the exponent x. So, when we write \(\log_a b = x\), we're simply finding the exponent x that makes \(a^x = b\) true. This operation is the bedrock of what logarithms are all about.
Exponentiation
Exponentiation is a mathematical operation where a number, known as the base, is raised to a power, which is the exponent. This process multiplies the base by itself as many times as indicated by the exponent.

For example, the expression \(2^3\) denotes 2 raised to the power of 3, which equals 2 脳 2 脳 2, or 8. Exponentiation is the counter operation to logarithms. While exponentiation helps find the total after repeatedly multiplying a number, logarithms help us backtrack to find the original exponent, embodying the concept of inverse operations.
Common Logarithm
The base-10 logarithm, commonly referred to as the common logarithm, signifies logarithms that use 10 as their base. The symbol \(\log b\) without a base usually implies that the base is 10, as in \(\log_{10} b\).

This particular logarithm is extensively used because our number system is decimal (base-10), which makes calculations with common logarithms intuitively easier to grasp and work with. In computing the common logarithm, you're essentially asking, 'To what power must 10 be raised to attain a specific number?' For example, the common logarithm of 1000 is 3 since \(10^3 = 1000\). This is denoted as \(\log 1000 = 3\).
Inverse Operations in Mathematics
Inverse operations are pairs of mathematical operations that reverse the effects of each other, such as addition and subtraction or multiplication and division. Logarithms and exponentiation are another pair of such inverse operations.

Understanding the relationship between a function and its inverse is crucial, as it enables the solving of equations by applying the inverse operation. For example, if you're given \(2^x = 8\), you can use the inverse operation, the logarithm, to solve for x, finding that \(x = \log_2 8\), which equals 3. This illustrates how logarithms serve as a bridge to simplify and solve exponential 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

When a strong acid is added to a buffer, the \(\mathrm{pH}\) changes a little bit. (a) Does the pH increase or decrease? (b) Why does the \(\mathrm{pH}\) change at all? Why doesn't the buffer hold the \(\mathrm{pH}\) constant?

It is possible to make two completely different buffers using the dihydrogen phosphate ion, \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) (a) In one buffer, \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) serves as the weak acid. What is the conjugate weak base? (b) Write the equations that show how the buffer in part (a) works when either \(\mathrm{H}_{3} \mathrm{O}^{+}\) or \(\mathrm{OH}^{-}\) is added. (c) In the other buffer, \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) serves as the weak base. What is the conjugate weak acid? (d) Write the equations that show how the buffer in part (c) works when either \(\mathrm{H}_{3} \mathrm{O}^{+}\) or \(\mathrm{OH}^{-}\) is added.

Amines are organic compounds that contain an \(\mathrm{NH}_{2}\) group, and water-soluble amines are weak bases in water. For example, the compound methylamine, \(\mathrm{H}_{3} \mathrm{C}-\mathrm{NH}_{2}\), is a weak base. (a) Draw a dot diagram for methylamine. (b) Using dot diagrams, show the equilibrium reaction between methylamine and water. (c) To which side does the equilibrium in part (b) lie? What did we tell you that allowed you to figure out the answer? (d) The similar compound ethane, \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{3}\), does not act as a weak base. Why can methylamine act as a weak base but \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{3}\) can't? (Hint: Draw a dot diagram for \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{3}\).) (e) Is it appropriate to call methylamine an electrolyte? If so, is it weak or strong? Explain.

Use the Bronsted-Lowry definition to explain why \(\mathrm{NH}_{3}\) is a base in water.

Is a solution that contains \(0.10 \mathrm{M} \mathrm{HBr}\) and \(0.10 \mathrm{M}\) NaBr a buffered solution? Why or why not?
See all solutions

Recommended explanations on Chemistry Textbooks

Inorganic Chemistry

Read Explanation

Making Measurements

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Chemistry Branches

Read Explanation

Physical Chemistry

Read Explanation

Organic Chemistry

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.