Problem 30
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. $$ 9 e^{x}=107 $$
Step-by-Step Solution
Verified Answer
The solution of the equation is \(x = \ln\left(\frac{107}{9}\right)\), which approximates to \(x \approx 0.99\).
1Step 1 - Isolate the exponential function
First, isolate the exponential term on one side of the equation by dividing both sides by 9. This gives: \[ e^{x}=\frac{107}{9} \]
2Step 2 - Apply the natural logarithm
Now apply the natural logarithm (with base \(e\)) on both sides of the equation. The base \(e\) of the natural logarithm and the base of the exponential function on the left-hand side are the same. So, applying ln to both sides of the equation gives: \[ \ln(e^{x})=\ln\left(\frac{107}{9}\right) \]
3Step 3 - Simplify the equation
The logarithm of an exponential (with the same base) is just the exponent. Thus this simplifies to: \[ x = \ln\left(\frac{107}{9}\right) \]
4Step 4 - Calculate the decimal approximation
Now, you can plug the right side of the equation into a calculator to get the decimal approximation: \[ x \approx 0.99 \]
Key Concepts
Natural LogarithmsExponential FunctionDecimal Approximation
Natural Logarithms
When delving into the realm of algebra, we come across various types of logarithms. Natural logarithms are one special type that are denoted as \( \ln \). The natural logarithm is the inverse operation to exponentiation with the special number \( e \), which is approximately equal to 2.71828. This number, \( e \), is known as Euler's number and is the base of the natural logarithm. This means that if you have \( \ln(e^x) \) the logarithm effectively 'cancels out' the exponentiation, leaving you with just \( x \).
When solving exponential equations like \( e^x = b \) where \( b \) is any real number, taking the natural logarithm of both sides allows us to isolate \( x \) and solve for it. In practice, this operation is commonly used in various scientific fields, including finance and biology, to model phenomena that grow at a constant rate per time period. This concept is critical in understanding how to manipulate and solve equations involving exponential functions.
When solving exponential equations like \( e^x = b \) where \( b \) is any real number, taking the natural logarithm of both sides allows us to isolate \( x \) and solve for it. In practice, this operation is commonly used in various scientific fields, including finance and biology, to model phenomena that grow at a constant rate per time period. This concept is critical in understanding how to manipulate and solve equations involving exponential functions.
Exponential Function
An exponential function is a mathematical function of the form \( y = a e^{bx} \) (where \( a \), \( b \) and \( x \) are constants and \( e \) is the base of the natural logarithm). The variable \( x \) is the exponent, hence the term 'exponential.' These functions are distinguished by their rate of growth or decay, which is proportional to the size of the function's current value. This unique growth pattern makes exponential functions relevant and ubiquitous in real-world applications, ranging from computing compound interest to analyzing radioactive decay.
The key aspect that you will often employ while solving an exponential equation is to isolate the term with \( e \) first. Once you have \( e^{x} = c \) where \( c \) is a constant, you'll apply logarithmic functions — typically the natural logarithm — to both sides. This transforms the equation into a form where the unknown exponent \( x \) can be more easily solved.
The key aspect that you will often employ while solving an exponential equation is to isolate the term with \( e \) first. Once you have \( e^{x} = c \) where \( c \) is a constant, you'll apply logarithmic functions — typically the natural logarithm — to both sides. This transforms the equation into a form where the unknown exponent \( x \) can be more easily solved.
Decimal Approximation
Getting to an exact solution for an equation involving natural logarithms or exponential functions often leads to results that are not 'nice' whole numbers. Instead, they are typically irrational numbers, which go on indefinitely without repeating. Therefore, it is practical to use a decimal approximation to express these numbers in a way that is easy to understand and use.
Decimal approximations are estimates of irrational numbers rounded to a certain number of decimal places. For instance, as seen in our textbook solution, \( \ln\left(\frac{107}{9}\right) \) is calculated and then rounded to two decimal places, giving us \( x \approx 0.99 \). This process involves using a calculator or computer to evaluate the natural logarithm to a desired level of precision. For most real-world applications, decimal approximations are more than sufficient and enable clear communication of the results derived from complex equations.
Decimal approximations are estimates of irrational numbers rounded to a certain number of decimal places. For instance, as seen in our textbook solution, \( \ln\left(\frac{107}{9}\right) \) is calculated and then rounded to two decimal places, giving us \( x \approx 0.99 \). This process involves using a calculator or computer to evaluate the natural logarithm to a desired level of precision. For most real-world applications, decimal approximations are more than sufficient and enable clear communication of the results derived from complex equations.
Other exercises in this chapter
Problem 30
Use the exponential decay model, \(A=A_{0} e^{k t},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of aspirin in your bloods
View solution Problem 30
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 30
In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{6} \sqrt{6} $$
View solution 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