/*! 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 10 Rewrite the following equations ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 10

Rewrite the following equations using logs instead of exponents. Estimate a solution for \(x\) and then check your estimate with a calculator. Round the value of \(x\) to three decimal places. a. \(10^{x}=153\) b. \(10^{x}=153,000\) c. \(10^{x}=0.125\) d. \(10^{x}=0.00125\)

Short Answer

Expert verified
a. \(x \approx 2.185\), b. \(x \approx 5.185\), c. \(x \approx -0.903\), d. \(x \approx -2.903\)

Step by step solution

01

Rewrite Equation a

Given the equation is \(10^{x}=153\). We need to rewrite it using logarithms. Take the common logarithm (base 10) on both sides: \(\log(10^{x})=\log(153)\)
02

Simplify Using Logarithm Properties for Equation a

Using the logarithm property, \(\log(a^b) = b \log(a)\), we get: \(x \cdot \log(10)=\log(153)\). Since \(\log(10) = 1\), we have: \(x = \log(153)\)
03

Estimate Solution Using Calculator for Equation a

Using a calculator, evaluate \(\log(153)\) to get: \(x \approx 2.185\)
04

Rewrite Equation b

Given the equation is \(10^{x}=153,000\). Take the common logarithm (base 10) on both sides: \(\log(10^{x})=\log(153,000)\)
05

Simplify Using Logarithm Properties for Equation b

Using the logarithm property, \(\log(a^b) = b \log(a)\), we get: \(x \cdot \log(10)=\log(153,000)\). Since \(\log(10) = 1\), we have: \(x = \log(153,000)\)
06

Estimate Solution Using Calculator for Equation b

Using a calculator, evaluate \(\log(153,000)\) to get: \(x \approx 5.185\)
07

Rewrite Equation c

Given the equation is \(10^{x}=0.125\). Take the common logarithm (base 10) on both sides: \(\log(10^{x})=\log(0.125)\)
08

Simplify Using Logarithm Properties for Equation c

Using the logarithm property, \(\log(a^b) = b \log(a)\), we get: \(x \cdot \log(10)=\log(0.125)\). Since \(\log(10) = 1\), we have: \(x = \log(0.125)\)
09

Estimate Solution Using Calculator for Equation c

Using a calculator, evaluate \(\log(0.125)\) to get: \(x \approx -0.903\)
10

Rewrite Equation d

Given the equation is \(10^{x}=0.00125\). Take the common logarithm (base 10) on both sides: \(\log(10^{x})=\log(0.00125)\)
11

Simplify Using Logarithm Properties for Equation d

Using the logarithm property, \(\log(a^b) = b \log(a)\), we get: \(x \cdot \log(10)=\log(0.00125)\). Since \(\log(10) = 1\), we have: \(x = \log(0.00125)\)
12

Estimate Solution Using Calculator for Equation d

Using a calculator, evaluate \(\log(0.00125)\) to get: \(x \approx -2.903\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithms
Logarithms are a fundamental concept in algebra that help us solve for unknown exponents. Instead of asking what power we need to raise a base to get a certain number, a logarithm provides us with the exponent directly. The logarithm of a number is the exponent to which the base must be raised to yield that number. In this exercise, common logarithms (base 10) are used. This is often written as \(\text{log} \). For instance, \( \text{log}_{10}(153) \) asks us what power of 10 gives us 153.
Exponential Equations
Exponential equations feature variables in the exponent, making them distinct from other algebraic equations. For example, \( 10^{x} = 153 \) is an exponential equation as the variable x is the exponent. The key challenge is to solve for x. To do this, we convert the exponential equation into its logarithmic form, where the variable is no longer an exponent but an operand of the logarithm. This conversion simplifies the solving process.
Algebraic Properties
Certain algebraic properties of logarithms are essential when solving logarithmic equations. The most useful properties in this scenario are:
  • The Power Rule: \( \text{log}_{b}(a^c) = c * \text{log}_{b}(a) \)
  • The Product Rule: \( \text{log}_{b}(mn) = \text{log}_{b}(m) + \text{log}_{b}(n) \)
  • The Quotient Rule: \( \text{log}_{b}(m/n) = \text{log}_{b}(m) - \text{log}_{b}(n) \)
These properties allow us to manipulate and simplify logarithmic expressions, making it easier to solve equations.
Calculator Usage
Once we have rewritten our exponential equations into logarithmic form, a calculator becomes invaluable. Calculators have built-in functions to compute logarithms quickly. For instance, you can use the \(\text{log}\) button, typically found on any scientific calculator, to find \( \text{log}(153) \) or \( \text{log}(0.125) \).
Here's a step-by-step on how to use your calculator for this:
  • Press the \( \text{log} \) button.
  • Enter the number you are taking the logarithm of.
  • Press the \( = \) button to get the result.
Always remember to round your answers to three decimal places, as specified in the exercise.

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Most popular questions from this chapter

In 2006 the United Kingdom generated approximately 81 terawatt-hours of nuclear energy for a population of about 60.6 million on 94,525 miles \(^{2}\). In the same year the United States generated approximately 780 terawatt-hours of nuclear energy for a population of about 300 million on 3,675,031 miles \(^{2}\). A terawatt is \(10^{12}\) watts. a. How many terawatt-hours is the United Kingdom generating per person? How many terawatt-hours is it generating per square mile? Express each in scientific notation. b. How many terawatt-hours is the United States generating per person? How many terawatt-hours are we generating per square mile? Express each in scientific notation. c. How much nuclear energy is being generated in the United Kingdom per square mile relative to the United States? d. Write a brief statement comparing the relative magnitude of generation of nuclear power per person in the United Kingdom and the United States.

Determine the value of each expression. a. \(|-5-3|\) c. \(|2-6|\) b. \(|6-2|\) d. \(-2|-1+3|+|-5|\)

Simplify where possible. Express your answer with positive exponents. a. \(\left(3 \cdot 3^{8}\right)^{-2}\) b. \(x^{3} \cdot x^{-4} \cdot x^{12}\) c. \(2^{6}+2^{6}+2^{7}+2^{-4}\) d. \(2 x^{-3}+3 x\left(x^{-4}\right)\) e. \(10^{-5}+5^{-2}+10^{10}\)

Compare the times listed below by plotting them on the same order-of-magnitude scale. (Hint: Start by converting all the times to seconds.) a. The time of one heartbeat ( 1 second) b. Time to walk from one class to another ( 10 minutes) c. Time to drive across the country ( 7 days) d. One year (365 days) e. Time for light to travel to the center of the Milky Way \((38,000\) years \()\) f. Time for light to travel to Andromeda, the nearest large galaxy ( 2.2 million years)

A TV signal traveling at the speed of light takes about \(8 \cdot 10^{-5}\) second to travel 15 miles. How long would it take the signal to travel a distance of 3000 miles?
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