Problem 31
Question
Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 3 e^{5 x}=1977 $$
Step-by-Step Solution
Verified Answer
The decimal equivalent of the solution to the equation \(3 e^{5 x}=1977\) is approximately \(x = 1.02\).
1Step 1: Isolate the exponential term.
The original equation is \(3e^{5x} = 1977\). First, divide both sides by 3 to isolate the exponential term on the left side. This will result in \(e^{5x} = 659\).
2Step 2: Apply the natural logarithm function.
Take the natural logarithm on both sides to get \(ln(e^{5x}) = ln(659)\). By using the property of logarithms that allows bringing down the exponents, the left side of the equation simplifies to \(5x*ln(e) = ln(659)\). Since the natural logarithm of \(e\) is 1, the equation simplifies further to \(5x = ln(659)\).
3Step 3: Solve for the variable x.
To isolate \(x\), divide both sides by 5. So, \(x =\frac{ln(659)}{5}\). This is the exact solution in terms of natural logarithms.
4Step 4: Approximate the solution.
To get the decimal approximation, use a calculator to find the value of \(\frac{ln(659)}{5}\). Round the result to two decimal places.
Key Concepts
Natural LogarithmsProperties of LogarithmsExponential Functions
Natural Logarithms
Understanding natural logarithms is crucial when dealing with exponential equations. A natural logarithm, denoted as \(\ln(x)\), is essentially the logarithm with the base of Euler's number, \(e\), which is an irrational constant approximately equal to 2.71828. When we say \(\ln(e)\), we are asking, 'to what power must we raise \(e\) to get \(e\)?' The answer is 1 because any number raised to the power of 1 is itself.
When solving exponential equations, taking the natural logarithm of both sides is a common technique used to isolate the variable, as seen in our exercise. This process utilizes a key property—that the natural logarithm of \(e^x\) is \(x\). So, when you come across \(e^x\), you can take the natural logarithm \(\ln\) of both sides, and the exponential expression simplifies to \(x\), allowing you to solve for the variable.
When solving exponential equations, taking the natural logarithm of both sides is a common technique used to isolate the variable, as seen in our exercise. This process utilizes a key property—that the natural logarithm of \(e^x\) is \(x\). So, when you come across \(e^x\), you can take the natural logarithm \(\ln\) of both sides, and the exponential expression simplifies to \(x\), allowing you to solve for the variable.
Properties of Logarithms
Logarithms have special properties that make them invaluable in solving exponential equations. Among these, two primary ones were used in the exercise: the logarithm of a product and the power rule.
The power rule states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base: \(\ln(a^b) = b \ln(a)\). This rule enables us to move the exponent in front of the logarithm, greatly simplifying the process of solving for the variable in exponential equations.
Another useful property is the fact that the natural logarithm of \(e\), \(\ln(e)\), is equal to 1. This is because logarithmic and exponential functions are inverse operations—like addition and subtraction or multiplication and division. By understanding and applying these properties, students can solve complex-looking equations through a series of logical and straightforward steps.
The power rule states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base: \(\ln(a^b) = b \ln(a)\). This rule enables us to move the exponent in front of the logarithm, greatly simplifying the process of solving for the variable in exponential equations.
Another useful property is the fact that the natural logarithm of \(e\), \(\ln(e)\), is equal to 1. This is because logarithmic and exponential functions are inverse operations—like addition and subtraction or multiplication and division. By understanding and applying these properties, students can solve complex-looking equations through a series of logical and straightforward steps.
Exponential Functions
Exponential functions are mathematical expressions that represent growth or decay. They are denoted as \(y = a \cdot e^{bx}\), where \(e\) is Euler's number, and \(a\) and \(b\) are constants. In the context of the given exercise, we dealt with such a function, \(3e^{5x} = 1977\).
To solve an equation involving an exponential function, it's often necessary to isolate the term containing the exponent, as was done in Step 1 of our example. Once the exponential is isolated, taking the logarithm of both sides allows us to utilize the properties of logarithms to bring down exponents and ultimately solve for \(x\).
Exponential functions are pervasive in real-world scenarios, such as compounding interest, population growth, and radioactive decay. Thus, understanding how to manipulate and solve these types of equations is not only essential for academic purposes but also for interpreting and modeling various phenomena in sciences, finance, and other fields.
To solve an equation involving an exponential function, it's often necessary to isolate the term containing the exponent, as was done in Step 1 of our example. Once the exponential is isolated, taking the logarithm of both sides allows us to utilize the properties of logarithms to bring down exponents and ultimately solve for \(x\).
Exponential functions are pervasive in real-world scenarios, such as compounding interest, population growth, and radioactive decay. Thus, understanding how to manipulate and solve these types of equations is not only essential for academic purposes but also for interpreting and modeling various phenomena in sciences, finance, and other fields.
Other exercises in this chapter
Problem 30
begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes Use
View solution Problem 31
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calcula
View solution Problem 31
In Exercises 21–42, evaluate each expression without using a calculator. $$ \cdot \log _{2} \frac{1}{\sqrt{2}} $$
View solution Problem 31
begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes Use
View solution