/*! 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 93 Solve for \(x\). Give an approxi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 93

Solve for \(x\). Give an approximation to four decimal places. $$ \log 692+\log x=\log 3450 $$

Short Answer

Expert verified
x 鈮� 4.9855

Step by step solution

01

Combine the logarithms on the left side

Use the properties of logarithms to combine the two logs on the left side. The property \(\log a + \log b = \log(ab)\) can be applied here: \(\log 692 + \log x = \log (692x)\).
02

Set the equation equal

The equation can now be written as: \(\log (692x) = \log 3450\).
03

Exponentiate both sides

To remove the logarithms, exponentiate both sides of the equation using the base of the logarithm (which is 10): \(10^{\log(692x)} = 10^{\log 3450}\). This simplifies to \(692x = 3450\).
04

Solve for x

Isolate x by dividing both sides of the equation by 692: \( x = \frac{3450}{692} = 4.9855\) (rounded to four decimal places).

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.

Properties of Logarithms
To understand how to solve logarithmic equations effectively, it's important to grasp the properties of logarithms. These properties are mathematical rules that allow us to simplify and manipulate logarithmic expressions.
In the given problem, the property we used is: \(\log a + \log b = \log(ab)\). This is known as the Product Rule of logarithms.
By knowing this property, we combined \(\log 692\) and \(\log x\) into \(\log(692x)\), simplifying the equation.
Exponentiation
Exponentiation is the process of raising a number (the base) to a power. When working with logarithms, we use exponentiation to undo the logarithmic function.
In our problem, we had the equation \(\log(692x) = \log 3450\).
To eliminate the logarithms, we exponentiated both sides using the base of the logarithm, which is 10:
\[10^{\log(692x)} = 10^{\log 3450}\]
This step is crucial because it simplifies the equation to \(692x = 3450\), making it easier to solve.
Logarithm Simplification
Simplifying logarithms helps transform complex logarithmic expressions into simpler forms. This often involves using properties of logarithms or performing algebraic manipulations.
In our exercise, we simplified \(\log 692 + \log x\) to \(\log(692x)\) by applying the Product Rule.
This simplification transformed a more complex addition problem into a single logarithmic term, thus making it easier to solve the equation.
Solving for a Variable
To solve for a variable in an equation means to isolate that variable on one side of the equation. This often involves using algebraic operations such as addition, subtraction, multiplication, or division.
In this example, once we simplified and exponentiated the equation to \(692x = 3450\), we solved for \(x\) by dividing both sides by 692:
\[x = \frac{3450}{692}\]
We then calculated this division to find that \(x \approx 4.9855\), providing the answer to four decimal places.

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

Graph function and its inverse using the same set of axes. \(f(x)=x^{2}-1, x \leq 0\)

The percentage of Americans ages \(30-49\) who use social networking sites can be estimated by $$ s(t)=1.1+36 \ln t $$ where \(t\) is the number of years after 2005 Data: Pew Research Center a) What percentage of Americans ages \(30-49\) used social networking sites in \(2012 ?\) b) Graph the function. c) In what year will \(95 \%\) of Americans ages \(30-49\) use social networking sites? d) What is the domain of the function?

Simplify. \(i^{43}\)

The SenderBase "Security Network ranks e-mail volume using a logarithmic scale. The magnitude \(M\) of a network's daily e-mail volume is given by $$ M=\log \frac{v}{1.34} $$ where \(v\) is the number of e-mail messages sent each day. How many e-mail messages are sent each day by a network that has a magnitude of \(7.5 ?\) Data: forum.spamcop.net

Simplify. $$ \frac{\sqrt[5]{a^{4} y^{6}}}{\sqrt{a y}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and 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.