/*! 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 122 Explain how to solve an exponent... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 122

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Short Answer

Expert verified
The solution for \(x\) in the equation \(3^{x}=140\) is approximated at \(x = \frac{ln(140)}{ln(3)}\), which when calculated gives us the value of \(x\) approximately equal to 4.77.

Step by step solution

01

Identify the exponential equation

The given exponential equation is \(3^{x}=140\). In this equation, 3 is the base, x is the exponent, and 140 is the result of the exponential operation. The goal is to find the value of x.
02

Take the logarithm of both sides

Applying the logarithm to both sides can help us to isolate variable from exponent. We take natural logarithm (ln) of both sides of the equation. This gives us the equation: \(ln(3^{x}) = ln(140)\).
03

Use the properties of logarithms

We use the property of logarithms that allows us to move the exponent in front of the logarithm. This gives us the equation: \(x*ln(3) = ln(140)\).
04

Isolate the variable

We can now find the value of x by dividing both sides of the equation by \(ln(3)\). This gives us the equation: \(x = \frac{ln(140)}{ln(3)}\).
05

Solve the equation

Solving the equation on the calculator provides the value of x.

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Most popular questions from this chapter

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b} x^{7} $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.9 billion for \(2010 ?\)

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Hamachiphobia $$ \begin{array}{ccc} {} & {\text { Percentage }} & {\text { Percentage } W h o} \\ {} & {\text { Who Won't }} & {\text { Don't Approve of }} \\ {\text { Generation }} & {\text { Try Sushi }} & {\text { Marriage Equality }} \\\ {\text { Millennials }} & {42} & {36} \\ {\text { Gen } X} & {52} & {36} \\ {\text { Boomers }} & {60} & {49} \\ {\text { Silent/Greatest }} & {72} & {66} \\ {\text { Generation }} \end{array} $$

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In \(2006,\) Canada's population exceeded Uganda's by 4.9 million.

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: pH (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.
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