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Chapter 4: Problem 65

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln 2 x=2.4\)

Short Answer

Expert verified
The solution to the equation \(\ln 2x = 2.4\) is approximately \(x = 2.714\) when rounded to three decimal places.

Step by step solution

01

Transcribe to Exponential Form

To solve for \(x\), we first need to convert our logarithmic equation into an exponential one. This can be done by using the fact that \(\ln a = b\) is equivalent to \(e^b = a\). Applying this to the given equation \(\ln 2x = 2.4\) transforms it into \(e^{2.4} = 2x\).
02

Solve for \(x\)

We now have the equation \(e^{2.4} = 2x\). We can solve for \(x\) by dividing both sides of the equation by 2 to isolate \(x\). This gives us \(x = e^{2.4}/2\).
03

Compute the Value

Now that we've isolated \(x\), we can use a calculator to work out the value of \(e^{2.4}/2\). Calculating this gives \(x = 5.48/2\), which simplifies to approximately \(x = 2.714\) (rounded to three decimal places).

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Key Concepts

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

From Logarithmic to Exponential Form
Understanding how to switch between logarithmic and exponential forms is crucial when solving equations involving logarithms. The natural logarithm, denoted as \(\ln x\), is a logarithm to the base \(e\), where \(e\) is the mathematical constant approximately equal to 2.718. When you have an equation like \(\ln a = b\), this can be rewritten in its exponential form as \(e^b = a\). This relationship exists because the natural logarithm is the power to which \(e\) must be raised to get the number \(a\).

For example, in the exercise provided, \(\ln 2x = 2.4\) is equivalent to \(e^{2.4} = 2x\) after conversion to exponential form. This step is key in the process of solving logarithmic equations because it transforms the problem into a more familiar algebraic equation.
Natural Logarithm Properties
The natural logarithm has unique properties that make it a valuable tool for solving equations. One such property is that the natural logarithm of \(e\), \(\ln e\), is always 1. This is because \(e\) raised to the power of 1 is \(e\). Another property is that the natural logarithm of 1, \(\ln 1\), is 0 because \(e\) raised to the power of 0 is 1. Moreover, the natural logarithm follows the rules of logarithms, such as:
  • Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)
  • Quotient Rule: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\)
  • Power Rule: \(\ln(a^b) = b\ln(a)\)

These properties are important for manipulating and simplifying logarithmic expressions when isolating variables and solving equations.
Isolating Variables Algebraically
Solving for an unknown variable often involves isolating it on one side of the equation. When dealing with exponential equations, such as those arising from logarithmic equations, isolating the variable requires specific algebraic operations. In the context of the exercise we are examining, once the logarithmic equation is converted into the exponential form (\(e^{2.4} = 2x\)), the next step is to perform algebraic operations that will leave \(x\) by itself on one side of the equation.

Dividing both sides by 2 (\(\frac{e^{2.4}}{2} = x\)) successfully isolates \(x\) and enables you to solve for it. The core concept here is to use inverse operations to undo the operations being performed on the variable. In the case of multiplication by 2, division by 2 is the inverse operation necessary to isolate \(x\). By mastering algebraic techniques to isolate variables, one can solve a wide range of mathematical problems.

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Most popular questions from this chapter

Stocking a Lake with Fish \(\quad\) A lake is stocked with 500 fish, and the fish population \(P\) increases according to the logistic curve \(P=\frac{10,000}{1+19 e^{-t / 5}}, \quad t \geq 0\) where \(t\) is the time (in months).

The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.

Domestic Demand The domestic demands \(D\) (in thousands of barrels) for refined oil products in the United States from 1995 to 2005 are shown in the table. (Source: U.S. Energy Information Administration)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Demand } \\ \hline 1995 & 6,469,625 \\ \hline 1996 & 6,701,094 \\ \hline 1997 & 6,796,300 \\ \hline 1998 & 6,904,705 \\ \hline 1999 & 7,124,435 \\ \hline 2000 & 7,210,566 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Demand } \\ \hline 2001 & 7,171,885 \\ \hline 2002 & 7,212,765 \\ \hline 2003 & 7,312,410 \\ \hline 2004 & 7,587,546 \\ \hline 2005 & 7,539,440 \\ \hline \end{array} $$(a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=5\) corresponding to 1995 . (b) Use the regression feature of a spreadsheet software program to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use the regression feature of a spreadsheet software program to find a logarithmic model \((y=a+b \ln x)\) for the data. (d) Use a spreadsheet software program to graph the exponential model in base \(e\) and the logarithmic model with the scatter plot. (e) Use both models to predict domestic demands in 2008 , 2009, and \(2010 .\) Do both models give reasonable predictions? Explain.

Super Bowl Ad Revenue The table shows Super Bowl TV ad revenues \(R\) (in millions of dollars) for several years from 1987 to 2006. (Source: TNS Media Intelligence)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Revenue } \\ \hline 1987 & 31.5 \\ \hline 1992 & 48.2 \\ \hline 1997 & 72.2 \\ \hline 2002 & 134.2 \\ \hline 2006 & 162.5 \\ \hline \end{array} $$(a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to 1987 . (b) Use the regression feature of a spreadsheet software program to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use a spreadsheet software program to graph the exponential model in base \(e\). (d) Use the exponential model in base \(e\) to predict the Super Bowl ad revenues in 2009 and in 2010 .

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.10}{12}\right)^{12 t}=2\)
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