/*! 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 57 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 57

Determine whether each statement is true or false. $$\log _{6} 60-\log _{6} 10=1$$

Short Answer

Expert verified
True

Step by step solution

01

- Apply the Logarithm Subtraction Rule

Use the rule \(\text{log}_a{b} - \text{log}_a{c} = \text{log}_a{\frac{b}{c}}\) to combine the logarithms: \(\text{log}_6{60} - \text{log}_6{10} = \text{log}_6{\frac{60}{10}}\).
02

- Simplify the Expression Inside the Logarithm

Simplify the fraction inside the logarithm: \(\frac{60}{10} = 6\). So now we have \(\text{log}_6{6}\).
03

- Evaluate the Logarithm

Evaluate \(\text{log}_6{6}\), which equals 1 because any logarithm of a number to its own base is equal to 1. Therefore, \(\text{log}_6{60} - \text{log}_6{10} = 1\).
04

- Determine the Truth Value

Since \(\text{log}_6{60} - \text{log}_6{10}\) simplifies correctly to 1, the given statement is true.

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 Subtraction Rule
The logarithm subtraction rule is a powerful property of logarithms. It allows you to subtract one logarithm from another easily. This rule states that for the same base, the difference of two logarithms is the logarithm of the division of their arguments. In mathematical terms, this is written as \(\text{log}_a{b} - \text{log}_a{c} = \text{log}_a{\frac{b}{c}}\).

This rule helps simplify complex expressions and consolidates multiple logarithms into one.

For example, in the given exercise, we used this rule to combine \(\text{log}_6{60} - \text{log}_6{10}\) into \(\text{log}_6{\frac{60}{10}}\), making further calculations more straightforward.
Simplify Logarithmic Expressions
Simplifying logarithmic expressions is a crucial skill. It involves using various properties of logarithms to make the expressions more manageable.

After applying the logarithm subtraction rule, our expression became \(\text{log}_6{\frac{60}{10}}\). Simplifying the fraction inside the logarithm is our next step.

In this case, \(\frac{60}{10} = 6\), leading to the expression \(\text{log}_6{6}\).

By simplifying logarithmic expressions, we can more easily see patterns or evaluate the expressions. Always look for ways to simplify fractions and combine multiple logarithms into a single term when possible.
Evaluate Logarithms
Evaluating logarithms means finding the actual number that a logarithm represents. The basic idea is to determine the power you need to raise the base to get the given number.

For instance, \(\text{log}_6{6}\) asks us: 'To what power must 6 be raised to get 6?' The answer is 1, so \(\text{log}_6{6} = 1\).

This is a general property of logarithms: \(\text{log}_a{a} = 1\). Any number's logarithm to its own base will always equal 1, as a number raised to the power of 1 is itself.

In our problem, once we simplified to \(\text{log}_6{6}\), we could easily evaluate it as 1, confirming that the original statement \(\text{log}_6{60} - \text{log}_6{10} = 1\) is true.

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

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\left(\log _{a} p-\log _{a} q\right)+2 \log _{a} r$$

Solve each equation. \(x=\log _{27} 3\)

Suppose that you are an agent for a detective agency. Today's encoding function is \(f(x)=4 x-5 .\) Find the rule for \(f^{-1}\) algebraically.

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \sqrt{\frac{x y}{5}}$$

Solve each equation. \(\log _{x} 5=\frac{1}{2}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied 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.