/*! 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 27 Simplify. $$ \log _{10} 0.01... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 27

Simplify. $$ \log _{10} 0.01 $$

Short Answer

Expert verified
-2

Step by step solution

01

- Understand the Logarithm

The logarithm \(\text{log}_{10} 0.01\) asks the question: To what power must 10 be raised to produce 0.01?
02

- Convert to Exponential Form

Rewrite 0.01 as a power of 10: \(\text{log}_{10} 0.01\) can be written as \(\text{log}_{10} 10^{-2}\). This is because 0.01 is equal to \(10^{-2}\).
03

- Apply the Logarithm Property

Use the property of logarithms that states \(\text{log}_{a} a^b = b\). Applying this property, we get \(\text{log}_{10} 10^{-2} = -2\).

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.

Exponential Form
To understand logarithms, first understand exponential form. The equation \(10^{-2} = 0.01\) is in exponential form. It shows that when you raise 10 to the power of -2, you get 0.01. Exponential forms are useful because they help us rewrite numbers in a simpler way. They also make it easier to use logarithms.
For example:
  • \(10^1 = 10\)

  • \(10^2 = 100\)

  • \(10^{-2} = 0.01\)
Understanding these forms helps when converting logarithms to exponential equations.
Logarithm Property
Logarithms have properties that make them useful for solving equations. One important property is \(\text{log}_{a} a^b = b\). This means that if you take the logarithm of a number raised to an exponent, the result is just the exponent.
For example:
  • \(\text{log}_{10} 10^3 = 3\)

  • \(\text{log}_{2} 2^4 = 4\)
In our problem: \(\text{log}_{10} 10^{-2} = -2\). This property helps simplify logarithms and solve problems more easily.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. If you have an exponential function \(y = a^x\), its logarithmic function would be \(x = \text{log}_{a}y\). This means that logarithms help to undo the exponential function.
For example:
  • If \(y = 10^2\), then \(2 = \text{log}_{10} y\).

  • If \(y = 2^3\), then \(3 = \text{log}_{2} y\).
Understanding logarithmic functions helps in various fields like science, engineering, and finance.
They're very useful for simplifying complex problems.

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

Is it easier to find \(x\) given \(x=\log _{9} \frac{1}{3}\) or given \(9^{x}=\frac{1}{3} ?\) Explain your reasoning.

Solve. $$ \log _{10}\left(x^{2}+21 x\right)=2 $$

Find the domain of each function. $$g(x)=\left|x^{2}-6 x+7\right|$$

The supply and demand for the sale of stereos by Sound Ideas are given by $$ S(x)=e^{x} \quad \text { and } \quad D(x)=162,755 e^{-x} $$ where \(S(x)\) is the price at which the company is willing to supply \(x\) stereos and \(D(x)\) is the demand price for a quantity of \(x\) stereos. Find the equilibrium point.

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{2}-4\right)-\log _{a}(x+2)$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

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.