/*! 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 69 Use the special properties of lo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 69

Use the special properties of logarithms to evaluate each expression. $$\log _{5} 1$$

Short Answer

Expert verified
Using the property \(\log_{b}(1) = 0\), we get \(\log_{5}(1) = 0\).

Step by step solution

01

Recall the definition of logarithms

A logarithm \(\log_{b}(a)\) is the exponent to which the base \(b\) must be raised to produce the number \(a\). In other words, if \(b^c = a\), then \(\log_{b}(a) = c\).
02

Apply the property of logarithms for base-5

Using the property \(\log_{b}(1) = 0\), regardless of the base \(b\), since any base raised to the power of 0 is 1. Therefore, \(b^0 = 1\).
03

Evaluate the given expression

Using the property \(\log_{b}(1) = 0\), we get \(\log_{5}(1) = 0\).

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
Logarithms are a practical way to deal with exponential relationships. It is defined as the power to which a number, called the base, must be raised to produce another number. Mathematically, if we have a base, say, 'b', and it is raised to a power 'c' to get a number 'a', we write it as: $$ b^c = a $$. The logarithm is then written as $$ \log_{b}(a) = c $$ . Simply, logarithms answer the question: 'To what exponent must the base be raised, to yield this number?' It's the inverse of exponentiation. For example, if $$ 2^3 = 8 $$ then $$ \log_{2}(8) = 3 $$.
base
The base in a logarithm is the number that is raised to a power. In the expression $$ \log_{b}(a) $$, 'b' is the base. The base can be any positive number except 1. For example, in $$ \log_{10}(100) $$, the base is 10. It's important because it determines what function we're working with. Different bases can drastically change the value of a logarithm. Common bases include:
  • Base 10: Often called the common logarithm and written as $$ \log (a) $$.
  • Base 'e': Known as the natural logarithm and written as $$ \ln (a) $$, where e is an irrational number approximately equal to 2.718.
  • Base 2: Common in computer science, as it relates directly to binary representation.
When no base is written, it's typically assumed to be 10.
logarithm rules
Logarithms follow several useful properties or rules that make them easier to work with:

  • Product Rule: $$ \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) $$.
  • Quotient Rule: $$ \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) $$.
  • Power Rule: $$ \log_{b}(x^c) = c \cdot \log_{b}(x) $$.
  • Change of Base Formula: $$ \log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)} $$ for any positive base 'k'.
  • Log of 1: For any base 'b', $$ \log_{b}(1) = 0 $$ since any number raised to the power of 0 is 1.
These rules are invaluable for simplifying complex logarithmic expressions and solving logarithmic equations. For example, in the given exercise, knowing that $$ \log_{5}(1) = 0 $$ due to the rule above makes the solution straightforward and clear.

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. Give exact solutions. $$ \log _{3} x+\log _{3}(2 x+5)=1 $$

Find the amount of money in an account after 8 yr if \(\$ 4500\) is deposited at \(6 \%\) annual interest compounded as follows. (a) Annually (b) Semiannually (c) Quarterly (d) Daily (Use \(n=365 .)\) (e) Continuously

Use the special properties of logarithms to evaluate each expression. $$\log _{3} 27$$

Solve each equation. Give exact solutions. $$ \log _{2}(t+5)-\log _{2}(t-1)=\log _{2} 3 $$

Use the special properties of logarithms to evaluate each expression. $$\log _{4} \frac{1}{4}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.