Problem 8
Question
Solve each equation. Express all solutions in exact form. Do not use a calculator. $$7 \ln 2 x=10$$
Step-by-Step Solution
Verified Answer
The solution is \( x = \frac{e^{\frac{10}{7}}}{2} \).
1Step 1: Simplify the Equation
Start by dividing both sides of the equation by 7 to isolate the natural logarithm, i.e. \( \ln (2x) \). The equation becomes: \( \ln (2x) = \frac{10}{7} \).
2Step 2: Exponentiate Both Sides
Since we have a natural logarithm, we can eliminate the logarithm by exponentiating both sides of the equation with base \(e\). This gives us: \( 2x = e^{\frac{10}{7}} \).
3Step 3: Solve for x
Now, divide both sides by 2 to solve for \(x\). This results in: \( x = \frac{e^{\frac{10}{7}}}{2} \).
Key Concepts
Natural LogarithmsExponentiationExact Solutions
Natural Logarithms
Natural logarithms are a special kind of logarithm where the base is the irrational number, Euler's number, denoted as \( e \). The approximate value of \( e \) is 2.718, and it appears frequently in mathematics, especially when dealing with exponential growth or decay. Natural logarithms simplify complex calculations because they correspond to the inverse operation of exponentiation with base \( e \).
When you see \( \ln(x) \), it means you're asking: "To what power should \( e \) be raised to obtain \( x \)?" For example, \( \ln(e^2) = 2 \) because \( e \) raised to the power of 2 is \( e^2 \).
In the original exercise, the expression \( 7 \ln(2x) = 10 \) involves natural logarithms. The goal is to simplify the equation to an easier form by isolating \( \ln(2x) \). This requires dividing the entire equation by 7, leading to \( \ln(2x) = \frac{10}{7} \). Isolating the logarithm is necessary for applying inverse operations like exponentiation.
When you see \( \ln(x) \), it means you're asking: "To what power should \( e \) be raised to obtain \( x \)?" For example, \( \ln(e^2) = 2 \) because \( e \) raised to the power of 2 is \( e^2 \).
In the original exercise, the expression \( 7 \ln(2x) = 10 \) involves natural logarithms. The goal is to simplify the equation to an easier form by isolating \( \ln(2x) \). This requires dividing the entire equation by 7, leading to \( \ln(2x) = \frac{10}{7} \). Isolating the logarithm is necessary for applying inverse operations like exponentiation.
Exponentiation
Exponentiation is a mathematical operation that involves raising a number to a power. It's essentially repeated multiplication, where the number to be multiplied is the base, and the number of times to multiply it is the exponent.
When dealing with natural logarithms, exponentiation with base \( e \) is crucial because it neutralizes the logarithmic operation. This is done by applying the power of \( e \) to both sides of an equation that contains a natural logarithm.
In the step-by-step solution, after isolating the natural logarithm to get \( \ln(2x) = \frac{10}{7} \), you exponentiate both sides with base \( e \) to eliminate the logarithm. This transformation leads to \( 2x = e^{\frac{10}{7}} \). Exponentiation effectively 'undoes' the logarithm, allowing you to solve for the variable as it removes the logarithmic function from the equation.
When dealing with natural logarithms, exponentiation with base \( e \) is crucial because it neutralizes the logarithmic operation. This is done by applying the power of \( e \) to both sides of an equation that contains a natural logarithm.
In the step-by-step solution, after isolating the natural logarithm to get \( \ln(2x) = \frac{10}{7} \), you exponentiate both sides with base \( e \) to eliminate the logarithm. This transformation leads to \( 2x = e^{\frac{10}{7}} \). Exponentiation effectively 'undoes' the logarithm, allowing you to solve for the variable as it removes the logarithmic function from the equation.
Exact Solutions
In mathematics, an exact solution is an answer provided in its most reduced form, without estimations or approximate calculations such as decimal approximations. It represents the pure mathematical answer.
When solving equations involving logarithms and exponentials, especially without a calculator, finding an exact solution involves algebraically manipulating the equation to express the solution purely in terms of mathematical constants and expressions.
In our problem, the process concludes with the solution of \( x \) as \( x = \frac{e^{\frac{10}{7}}}{2} \). This is an exact form because it's written as a fraction of an exponential expression, showing precision and avoiding the use of decimal approximations, which can lose accuracy. This exact expression conveys the result of solving the equation authentically using mathematical operations.
When solving equations involving logarithms and exponentials, especially without a calculator, finding an exact solution involves algebraically manipulating the equation to express the solution purely in terms of mathematical constants and expressions.
In our problem, the process concludes with the solution of \( x \) as \( x = \frac{e^{\frac{10}{7}}}{2} \). This is an exact form because it's written as a fraction of an exponential expression, showing precision and avoiding the use of decimal approximations, which can lose accuracy. This exact expression conveys the result of solving the equation authentically using mathematical operations.
Other exercises in this chapter
Problem 8
Decide whether each function is one-to-one. $$f(x)=\sqrt[3]{x}$$
View solution Problem 8
For each statement, write an equivalent statement in logarithmic form. Do not use a calculator. $$e^{1 \Omega}=\sqrt[3]{e}$$
View solution Problem 8
Solve each equation. Do not use a calculator. $$\left(\frac{2}{3}\right)^{x}=\frac{9}{4}$$
View solution Problem 9
Decide whether each function is one-to-one. $$f(x)=|2 x+1|$$
View solution