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Chapter 5: Problem 44

Solve the equations. Express the answers in terms of natural logarithms. $$3 e^{1+t}=2$$

Short Answer

Expert verified
\( t = \ln\left(\frac{2}{3}\right) - 1 \)

Step by step solution

01

Isolate the exponential expression

First, start by dividing both sides of the equation by 3 to isolate the term involving the exponential function. This gives us:\[ e^{1+t} = \frac{2}{3} \]
02

Apply the natural logarithm

Now apply the natural logarithm (ln) to both sides of the equation to remove the base \(e\). This helps us solve for the variable inside the exponent:\[ \ln(e^{1+t}) = \ln\left(\frac{2}{3}\right) \]
03

Simplify using logarithm properties

Use the property of logarithms that states \( \ln(e^x) = x \). So the equation simplifies to:\[ 1 + t = \ln\left(\frac{2}{3}\right) \]
04

Solve for t

Finally, solve for \(t\) by subtracting 1 from both sides:\[ t = \ln\left(\frac{2}{3}\right) - 1 \]

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Key Concepts

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

Solving Equations
Solving equations is like solving a puzzle.
The goal is to find the value of the variable that makes the equation true. To solve the equation, we need to perform operations that simplify the equation step by step until the variable is isolated.
In our example, the equation involves an exponential term, \( e^{1+t} \). The plan is to isolate this term by performing the opposite operations.
  • First, divide both sides by 3 to simplify the situation. This keeps the equation balanced while focusing on the exponential part.
  • Our goal in solving equations like this is to simplify until we reach the variable itself. Here, we progress towards isolating \( t \) by handling the exponential function separately.
Exponential Functions
Exponential functions are special and powerful. They express growth and decay in nature, finance, and more. In mathematics, an exponential function with base \( e \) is uniquely important.
The equation \( e^{1+t} = \frac{2}{3} \) showcases this type of function.
  • Here, \( e \) is Euler's number, approximately 2.718, and serves as the base of natural logarithms.
  • Exponential functions feature a variable in the exponent. This makes them different from linear or polynomial functions.
  • To "undo" an exponential function, we apply logarithms, effectively switching focus from the exponential form to a more manageable linear expression.
Understanding exponential functions is key to solving related equations. Recognizing when and how to use these functions helps simplify and solve such problems.
Logarithmic Properties
Logarithmic properties are like the tools in a mathematician鈥檚 toolbox. When dealing with equations with exponential parts, logarithms are particularly handy. They help us get the variable out of the exponent.
  • The natural logarithm, \( \ln \), corresponds to the base \( e \). It acts as the inverse operation to raising \( e \) to a power.
  • One crucial property is that \( \ln(e^x) = x \). This means if you have a logarithm of an exponent with base \( e \), the logarithm simplifies directly to what the exponent was.
This property was crucial in our example. Applying \( \ln \) to both sides allowed us to easily handle the term \( e^{1+t} \) and solve for \( t \). Understanding these properties gives us the ability to tackle equations involving logarithms and exponentials with confidence.

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Most popular questions from this chapter

(a) Use a graphing utility to estimate the root(s) of the equation to the nearest one-tenth (as in Example 6). (b) Solve the given equation algebraically by first rewriting it in logarithmic form. Give two forms for each answer: an exact expression and a calculator approximation rounded to three decimal places. Check to see that each result is consistent with the graphical estimate obtained in part (a). $$e^{3 x^{2}}=112$$

Solve the inequalities. \(\log _{\pi}\left[\log _{4}\left(x^{2}-5\right)\right]<0 \quad\) Hint: In the expression \(\log _{b} y, y\) must be positive.

Solve each equation. $$\frac{\ln (\sqrt{x+4}+2)}{\ln \sqrt{x}}=2$$

Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$4^{5-x}>15$$

In the text we showed that the relative growth rate for the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) is constant for all time intervals of unit length, \([t, t+1] .\) Recall that we did this by computing the relative change \([\mathcal{N}(t+1)-\mathcal{N}(t)] / \mathcal{N}(t)\) and noting that the result was a constant, independent of \(t .\) (If you've completed the previous exercise, you've done this calculation for yourself.) Now consider a time interval of arbitrary length, \([t, t+d] .\) The relative change in the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) over this time interval is \([\mathcal{N}(t+d)-\mathcal{N}(t)] / \mathcal{N}(t) .\) Show that this quantity is a constant, independent of \(t .\) (The expression that you obtain for the constant will contain \(e\) and \(d\), but not \(t\) As a check on your work, replace \(d\) by 1 in the expression you obtain and make sure the result is the same as that in the text where we worked with intervals of length \(d=1 .)\)
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