/*! 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 22 Simplify. $$ \log _{7} 1 $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 22

Simplify. $$ \log _{7} 1 $$

Short Answer

Expert verified
\(\log _{7} 1 = 0\).

Step by step solution

01

Understand Logarithm Definition

Recall the definition of a logarithm. The logarithmic function \(\text{log}_{b}(x)\) asks the question: 'To what power must the base \(b\) be raised to achieve \(x\)?'
02

Apply Logarithm Definition

In this case, the exercise asks for \(\text{log}_{7}(1)\). The question now becomes: 'To what power must 7 be raised to get 1?'
03

Identify the Power

We know that any number raised to the power of 0 gives 1: \{ b^0 = 1\ for any \( b e 0\)}. Therefore, \(7^{0} = 1\).
04

Simplify

Based on the information above, \(\text{log}_{7}(1) = 0\).

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
The concept of a logarithm might seem tricky at first, but it's quite straightforward once you break it down. A logarithm essentially answers the question: 'To what exponent must we raise a given base to obtain a certain number?' This relationship is symbolized as \(\text{log}_{b}(x)\). Here鈥檚 an example to clarify: \( \text{log}_{10}(100) \) asks us to find the exponent that 10 must be raised to get 100. Since \(10^2 = 100 \), \( \text{log}_{10}(100) = 2 \). In our exercise, \( \text{log}_{7}(1) \), we鈥檙e looking for the power to which 7 must be raised to produce 1.
Logarithmic Function
A logarithmic function serves as the inverse of an exponential function. When you see \( \text{log}_{b}(x) \), you鈥檙e dealing with questions about exponents. Here is a breakdown of its components:
- The 'base' is the little subscript number. In \( \text{log}_{7}(1) \), 7 is the base.
- The argument is what you鈥檙e taking the log of. In this exercise, it鈥檚 1.
- The result is the exponent that the base is raised to get the argument.
Simply put, \( \text{log}_{b}(x) \) and \( b^{\text{log}_{b}(x)} = x \) are two faces of the same coin. So, if you know how to handle exponents, you can handle logarithms just as well.
Power of Zero
Understanding why \( \text{log}_{7}(1) \) evaluates to 0 comes down to the properties of exponents. Any nonzero number raised to the power of zero is 1. This is a fundamental rule of exponents, expressed as:
\[ b^0 = 1 \text{ for any } b e 0 \]
For our exercise, we can clarify this with:
- Start with the base, which is 7.
- Ask, 'To what power must 7 be raised to yield 1?'
- The answer is 0 because \( 7^0 = 1 \).
Therefore, \( \text{log}_{7}(1) = 0 \). This real-world scenario simplifies logarithms by turning abstract concepts into clear, memorable rules.

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

Simplify. $$ \frac{a^{15}}{a^{3}} $$

$$\text { If } \log _{a} x=2, \text { what is } \log _{a}(1 / x) ?$$

Solve. $$ \log _{32} x=\frac{2}{3} $$

As of February \(2010,\) Alex Rodriguez of the New York Yankees had the largest contract in sports history. As part of the 10 -year \(\$ 275\) -million deal, he will receive \(\$ 20\) million in \(2016 .\) How much money would need to be invested in 2008 at \(4 \%\) interest, compounded continuously, in order to have \(\$ 20\) million for Rodriguez in \(2016 ?\) (This is much like determining what \(\$ 20\) million in 2016 is worth in 2008 dollars.

Simplify. $$\log _{Q} Q^{-2}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.