/*! 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 26 Find the indicated value of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 26

Find the indicated value of the logarithmic functions. $$\log \left(10^{6}\right)$$

Short Answer

Expert verified
6

Step by step solution

01

Understand the Logarithmic Function

The logarithmic function \(\log_b(x)\) answers the question 'To what power must we raise b to get x?'. In this case, it鈥檚 \(\log(10^6)\).
02

Apply the Definition of Logarithms

According to the definition, \(\log_b(b^x) = x\). Here, the base is 10 and the exponent is 6, so \(\log(10^6) = 6\).

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.

Logarithms
Logarithms are mathematical functions that tell us the power to which we need to raise a number, called the base, to get another number. If you have a logarithm like \(\text{log}_b(x)\), it asks, 'what power should base \( b \) be raised to, to become \( x \)?' For example, \( \text{log}_2(8) \) asks 'To what power must 2 be raised to produce 8?' and the answer is 3 because \(2^3 = 8\). It's important to remember that logarithms and exponents are closely related. They are, in fact, inverse operations.
Exponents
Exponents are numbers that show how many times a base number is multiplied by itself. For example, in \( 2^3 \), 2 is the base and 3 is the exponent, meaning \( 2 \times 2 \times 2 = 8 \). When we look at logarithms, we're essentially looking for what exponent we need for a given base to reach a particular value. To solve \( \text{log}(10^6) \), you need to ask yourself: '10 raised to what power equals \$10^6\$?'. The answer is simple: 10 raised to the power of 6 gives you 10^6.
Base 10 Logarithms
Base 10 logarithms, also known as common logarithms, have a base of 10. They are typically written as \( \text{log}(x) \) without the base explicitly written, since base 10 is the default. For example, in \( \text{log}(10^6) \), we understand that the base is 10. The fundamental definition to remember is that \( \text{log}_{10}(10^x) = x \). So, when you see \( \text{log}(10^6) \), it simplifies directly to 6 because we're asking what power 10 needs to be raised to in order to equal 10^6.

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

Solve each equation. Find the exact solutions. $$\log _{x}\left(\frac{1}{9}\right)=-\frac{2}{3}$$

Finding Relationships Graph each of the following pairs of functions on the same screen of a graphing calculator. (Use the base-change formula to graph with bases other than 10 or e.) Explain how the functions in each pair are related. a. \(y_{1}=\log _{3}(\sqrt{3} x), y_{2}=0.5+\log _{3}(x)\) b. \(y_{1}=\log _{2}(1 / x), y_{2}=-\log _{2}(x)\) c. \(y_{1}=3^{x-1}, y_{2}=\log _{3}(x)+1\) d. \(y_{1}=3+2^{x-4}, y_{2}=\log _{2}(x-3)+4\)

Solving for Time Solve the formula $$R=P \frac{i}{1-(1+i)^{-n t}}$$ for \(t .\) Then use the result to find the time (to the nearest month) that it takes to pay off a loan of \(\$ 48,265\) at \(8 \frac{3}{4} \%\) APR compounded monthly with payments of \(\$ 700\) per month.

Find the approximate solution to each equation. Round to four decimal places. $$5 e^{x}=4$$

The cost of installing an oak floor varies jointly with the length and width of the room. If the cost is \(\$ 875.60\) for a room that is 8 feet by 11 feet, what is the cost for a room that is 10 feet by 14 feet?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.