/*! 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 4 Solve the exponential equation a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Solve the exponential equation algebraically. Then check using a graphing calculator. Round to three decimal places, if appropriate. $$3^{7 x}=27$$

Short Answer

Expert verified
x 鈮� 0.429

Step by step solution

01

Rewrite the Base

Rewrite the number 27 as a power of 3. Notice that 27 can be written as \(3^3\). So the equation becomes:\[3^{7x} = 3^3\]
02

Set the Exponents Equal

Since the bases are the same, we can set the exponents equal to each other:\[7x = 3\]
03

Solve for x

To solve for \(x\), divide both sides of the equation by 7:\[x = \frac{3}{7}\]
04

Round the Solution

Round the solution to three decimal places:\[x \approx 0.429\]
05

Verify Using a Graphing Calculator

Graph the functions \(y = 3^{7x}\) and \(y = 27\). The x-coordinate of the intersection point should be approximately 0.429, confirming our solution.

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.

algebraic solution
When solving exponential equations like the one given, we can use algebraic methods to find the solution. In this problem, we start by rewriting numbers to express them with the same base. For example, we can rewrite 27 as the power of 3, because 27 is equivalent to 3 raised to the power of 3, or simply \(27 = 3^3\).
  • This makes our original equation \(3^{7x} = 27\) become \(3^{7x} = 3^3\).

With the same base on both sides, we set the exponents equal to each other (\(7x = 3\)) and solve for \(x\) by dividing both sides of the equation by 7. This step gives us: \[7x = 3 \quad \Rightarrow \quad x = \frac{3}{7} = 0.428571...\] This fraction represents our solution, which simplifies algebraically to find the exact value of \(x\).
graphing calculator
Using a graphing calculator can help visualize and confirm the algebraic solution. First, input the functions into your graphing calculator:
  • \(y = 3^{7x}\)
  • \(y = 27\)
The graph will display two curves, and your task is to find their intersection point. Where these lines intersect, the x-coordinate should represent our solution for \(x\). To observe the intersection:
  • Zoom in on the region around the initial guess based on your algebraic solution, which is approximately 0.429.
  • Use the 'trace' function or cursor to precisely locate the intersection point.

Doing this, you should find the x-coordinate close to the algebraically calculated value of 0.429. This graphing method reinforces that our algebraic solution is accurate.
rounding decimals
Rounding decimals is crucial when working with irrational or long decimal numbers. Here's a quick guide to rounding to three decimal places:
  • Locate the third decimal place. For our fraction \(\frac{3}{7}\), it's 0.428571... so the third decimal place is 8.
  • Check the digit immediately after the third decimal place (the fourth decimal place). In this case, it's 5.

  • If the fourth digit is 5 or higher, round the third place up.
  • If the fourth digit is less than 5, leave the third place as it is.

Given our number is 0.428571..., and our fourth digit is 5, we round up, so the rounded decimal is 0.429. Rounding helps in simplifying numbers for practical use, without sacrificing too much precision in most cases.

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

Centenarian Population. The centenarian population in the United States has grown over \(65 \%\) in the last 30 years. In \(1980,\) there were only \(32,194\) residents ages 100 and over. This number had grown to \(53,364\) by \(2010 .\) (Sources: Population Projections Program; U.S. Census Bureau; U.S. Department of Commerce; "What People Who Live to 100 Have in Common," by Emily Brandon, U.S. News and World Report, January \(7,2013\) ) The exponential function $$ H(t)=80,040.68(1.0481)^{t} $$ where \(t\) is the number of years after \(2015,\) can be used to project the number of centenarians. Use this function to project the centenarian population in 2020 and in 2050 (IMAGE CANT COPY)

Sketch the graph of the function and check the graph with a graphing calculator. Describe how each graph can be obtained from the graph of a basic exponential function. $$y=e^{2 x}+1$$

The Beer-Lambert Law. A beam of light enters a medium such as water or smog with initial intensity \(I_{0} .\) Its intensity decreases depending on the thickness (or concentration) of the medium. The intensity \(I\) at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$, The constant \(\mu\) (the Greek letter "mu") is called the coefficient of absorption, and it varies with the medium. For sea water, \(\mu=1.4\) a) What percentage of light intensity \(I_{0}\) remains in sea water at a depth of \(1 \mathrm{m} ? 3 \mathrm{m} ? 5 \mathrm{m} ?\) \(50 \mathrm{m} ?\) b) Plant life cannot exist below \(10 \mathrm{m}\). What percentage of \(I_{0}\) remains at \(10 \mathrm{m} ?\)

Compound Interest. Suppose that \(\$ 750\) is invested at \(7 \%\) interest, compounded semiannually. a) Find the function for the amount to which the investment grows after \(t\) years. b) Graph the function. c) Find the amount of money in the account at \(t=1,6\) \(10,15,\) and 25 years. d) When will the amount of money in the account reach \(\$ 3000 ?\)

Newton's Law of Cooling. Suppose that a body with temperature \(T_{1}\) is placed in surroundings with temperature \(T_{0}\) different from that of \(T_{1}\). The body will either cool or warm to temperature \(T(t)\) after time \(t,\) in minutes, where $$T(t)=T_{0}+\left(T_{1}-T_{0}\right) e^{-i t}$$,A chilled gelatin salad that has a temperature of \(43^{\circ} \mathrm{F}\) is taken from the refrigerator and placed on the dining room table in a room that is \(68^{\circ} \mathrm{F}\). After \(12 \mathrm{min}\), the temperature of the salad is \(55^{\circ} \mathrm{F}\). What will the temperature of the salad be after 20 min?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.