/*! 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 35 Evaluate each expression without... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 35

Evaluate each expression without using a calculator. $$\log _{5} 5$$

Short Answer

Expert verified
The answer is 1.

Step by step solution

01

Identifying the Problem

The task involves finding the value of \( \log _{5} 5\). This notation is to be interpreted as '5 to the power of what equals 5'.
02

Applying Logarithm Rules

Recall the logarithm property that for any base \(b\), \(\log _{b} b = 1\). This applies because any number to the power of 1 equals itself.
03

Final Answer

Hence, applying this property to the problem at hand, we find that \(\log _{5} 5 = 1\).

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.

Understanding Logarithm Properties
Logarithms have unique properties that make them a powerful mathematical tool. One such property is the identity: for any positive number, the logarithm of a number to its own base is 1.
This is represented as \( \log_{b} b = 1 \). The principle behind this is simple. Any number raised to the power of 1 equals the number itself.
This concept helps us simplify expressions and solve equations involving logarithms.
  • For example, \( \log_{10} 10 = 1 \).
  • Similarly, \( \log_{e} e = 1 \) (where \( e \) is the base of natural logarithms).
Understanding these basic properties is crucial in solving more complicated math problems involving logarithms.
Techniques for Evaluating Logarithms
Evaluating logarithms can seem daunting at first. However, if you understand the properties of logarithms, it becomes much simpler.
The key is to interpret the logarithm correctly: it asks the question, "What exponent do we raise our base to in order to get a particular number?"
In the example of \( \log_{5} 5 \), we're considering what power of 5 gives us 5.
Using the property \( \log_{b} b = 1 \), we see that the exponent is 1. Another useful technique when evaluating logarithms without a calculator is to know that:
  • If the number is the same as the base, the logarithm is always 1.
  • If the number is a power of the base, the logarithm equals that power.
  • If the number is a root of the base, the logarithm will be a fractional number.
Mathematics Problem Solving with Logarithms
When solving problems involving logarithms, it's essential to apply logical reasoning and logarithm rules elegantly.
To solve the original exercise \( \log_{5} 5 \), the task is first to identify the base and the number you're working with. Knowing the properties makes this process smoother.
Here's a friendly breakdown to problem solve with logarithms:
  • Clarify what the logarithm is asking (identify base, number).
  • Apply the property \( \log_{b} b = 1 \) to evaluate the expression efficiently.
  • Practice makes perfect 鈥� the more you practice, the more intuitive these problems will become.
Alongside these steps, always check your work: Does the answer make sense? Is it logically consistent with what we know about logarithms?

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

Make Sense? In Exercises \(137-140,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

This will help you prepare for the material covered in the first section of the next chapter. $$\text { Solve: } \frac{5 \pi}{4}=2 \pi x$$

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

Use a calculator with an \(\left[e^{x}\right]\) key to solve. The bar graph shows the percentage of U.S. high school seniors who applied to more than three colleges for selected years from 1980 through 2013. (BAR GRAPH CAN'T COPY) The data can be modeled by $$ f(x)=x+31 \text { and } g(x)=32.7 e^{0.0217 x} $$ in which \(f(x)\) and \(g(x)\) represent the percentage of high school seniors who applied to more than three colleges \(x\) years after 1980\. Use these functions to solve . Where necessary, round answers to the nearest percent. In college, we study large volumes of information \(-\) information that, unfortunately, we do not often retain for very long. The function $$ f(x)=80 e^{-0.5 x}+20 $$ describes the percentage of information, \(f(x),\) that a particular person remembers \(x\) weeks after learning the information. a. Substitute 0 for \(x\) and, without using a calculator, find the percentage of information remembered at the moment it is first learned. b. Substitute 1 for \(x\) and find the percentage of information that is remembered after 1 week. c. Find the percentage of information that is remembered after 4 weeks. d. Find the percentage of information that is remembered after one year (52 weeks).

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied 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.