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Chapter 1: Problem 7

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$50=10 \cdot 3^{t}$$

Short Answer

Expert verified
The solution for \(t\) is approximately 1.4649.

Step by step solution

01

Isolate the Exponential Expression

Start with the equation: \[50 = 10 \cdot 3^t\]To isolate the exponential expression, divide both sides by 10:\[\frac{50}{10} = 3^t\]This simplifies to:\[5 = 3^t\]
02

Apply Natural Logarithm

Take the natural logarithm on both sides of the equation:\[\ln(5) = \ln(3^t)\]Using the logarithmic property \(\ln(a^b) = b\ln(a)\), we can rewrite the equation as:\[\ln(5) = t \cdot \ln(3)\]
03

Solve for t

To find the value of \(t\), divide both sides by \(\ln(3)\):\[t = \frac{\ln(5)}{\ln(3)}\]Calculating using a calculator gives:\[t \approx \frac{1.6094}{1.0986} \approx 1.4649\]

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Key Concepts

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

Exponential Equations
Exponential equations are central in many areas of mathematics and everyday applications, like population growth or radioactive decay. In an exponential equation, the unknown variable is found in the exponent, as shown in the exercise equation: \[50 = 10 \cdot 3^t\] Here, \(t\) is the exponent we need to solve for. A key strategy when dealing with exponential equations is to isolate the exponential part by performing operations that simplify it.
  • Divide or multiply both sides to isolate the exponential expression.
  • In this exercise, dividing by 10 simplifies things: \[5 = 3^t\]
Once isolated, you are ready to transform the expression using logarithms, a powerful tool in solving exponential equations.
Solving Logarithmic Equations
Solving logarithmic equations often involves applying logarithms to handle the exponent, as seen with the application of the natural logarithm in the exercise. By applying the natural logarithm (\(\ln\)), you capitalize on its properties to linearize the equation:\[\ln(5) = \ln(3^t)\]Using the property of logarithms, \(\ln(a^b) = b \ln(a)\), simplifies this to:\[\ln(5) = t \cdot \ln(3)\]
  • The power of the expression becomes a coefficient through logarithmic properties, helping transform a multiplicative relationship into a linear one.
  • In this specific case, natural logarithms are optimal because they pair well with exponential functions. Specifically, those involving base \(e\), but they are broadly useful as shown here with base 3.
After reaching a linear form, solving for the unknown resembles solving any typical algebraic equation.
Mathematics Problem-Solving
Mathematics problem-solving often requires a strategic movement through steps that build upon each part systematically. In this scenario, we addressed a problem using a methodical approach to find \(t\). First, isolate the variable, then transform the equation using effective tools like logarithms, and finally solve for the unknown.This exercise highlights key steps including:
  • Ensuring the equation is in a suitable form to apply further mathematical operations.
  • Recognizing which mathematical tool, like natural logarithms, best suits the problem.
  • Applying algebraic techniques, i.e., simplifying using log properties, to achieve a solvable linear equation.
By following these techniques, solving complex equations becomes manageable, cultivating a deeper understanding and developing strong problem-solving skills.

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Most popular questions from this chapter

Soybean production, in millions of tons $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Production } & 161.0 & 170.3 & 180.2 & 190.7 & 201.8 & 213.5 \\ \hline \end{array}$$ Aircraft require longer takeoff distances, called takeoff rolls, at high altitude airports because of diminished air density. The table shows how the takeoff roll for a certain light airplane depends on the airport elevation. (Takeoff rolls are also strongly influenced by air temperature; the data shown assume a temperature of \(0^{\circ} \mathrm{C}\) ) Determine a formula for this particular aircraft that gives the takeoff roll as an exponential function of airport elevation. $$\begin{array}{c|c|c|c|c|c} \hline \text { Elevation (ft) } & \text { Sca level } & 1000 & 2000 & 3000 & 4000 \\ \hline \text { Takeoff roll (ft) } & 670 & 734 & 805 & 882 & 967 \\ \hline \end{array}$$

A quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{b} a^{t}\) to: (a) Find values for the parameters \(a\) and \(P_{0}\). (b) State the initial quantity and the percent rate of growth or decay. \(P=320\) when \(t=5\) and \(P=500\) when \(t=3\)

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In Problems \(25-28,\) put the functions in the form \(P=P_{0} e^{k t}\). $$P=4(0.55)^{t}$$

Values of a function are given in the following table. Explain why this function appears to be periodic. Approximately what are the period and amplitude of the function? Assuming that the function is periodic, estimate its value at \(t=15,\) at \(t=75,\) and at \(t=135\). $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline t & 20 & 25 & 30 & 35 & 40 & 45 & 50 & 55 & 60 \\\\\hline f(t) & 1.8 & 1.4 & 1.7 & 2.3 & 2.0 & 1.8 & 1.4 & 1.7 & 2.3 \\\\\hline\end{array}$$
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