Problem 113
Question
Use any method (analytic or graphical) to solve each equation. $$\ln \left(\ln e^{-x}\right)=\ln 3$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \( x = -3 \).
1Step 1: Rewriting the Equation
Start by rewriting the given equation in a more approachable form. The original equation is \( \ln(\ln e^{-x}) = \ln 3 \). Since \( \ln b = \ln c \) implies that \( b = c \), we have \( \ln e^{-x} = 3 \).
2Step 2: Solving for \( e^{-x} \)
Now, raise the base of the natural logarithm \( e \) to both sides of the equation to remove the natural log: \( e^{\ln e^{-x}} = e^3 \). This simplifies to \( e^{-x} = e^3 \).
3Step 3: Equating Exponents
Since the bases \( e \) are the same, equate the exponents: \(-x = 3 \).
4Step 4: Solving for \( x \)
Solve for \( x \) by multiplying both sides of the equation by \(-1\), giving \( x = -3 \).
Key Concepts
Natural LogarithmExponential FunctionsEquation SolvingAnalytic Methods
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a special type of logarithm that uses Euler's number, \( e \), as its base. The value of \( e \) is approximately 2.71828, and it is an irrational number crucial in mathematical analysis and calculus. Natural logarithms are particularly useful because they transform exponential growth into linear growth, making complex calculations more manageable.
When you see an expression like \( \ln(b) \), it tells you the power to which you need to raise \( e \) to get \( b \). That is, if \( \ln(b) = c \), then \( e^c = b \). This property is incredibly useful in algebra because it lets us "unwrap" the logarithm and solve equations. For instance, in our exercise, the equation \( \ln(\ln e^{-x}) = \ln 3 \) simplifies to \( \ln e^{-x} = 3 \) based on the principle that if \( \ln b = \ln c \), then \( b = c \). By applying the natural logarithm, we can manipulate exponential expressions into linear equations, which are generally easier to solve.
When you see an expression like \( \ln(b) \), it tells you the power to which you need to raise \( e \) to get \( b \). That is, if \( \ln(b) = c \), then \( e^c = b \). This property is incredibly useful in algebra because it lets us "unwrap" the logarithm and solve equations. For instance, in our exercise, the equation \( \ln(\ln e^{-x}) = \ln 3 \) simplifies to \( \ln e^{-x} = 3 \) based on the principle that if \( \ln b = \ln c \), then \( b = c \). By applying the natural logarithm, we can manipulate exponential expressions into linear equations, which are generally easier to solve.
Exponential Functions
Exponential functions are defined by the equation \( f(x) = a^x \), where \( a \) is a positive constant. In our discussion, the focus is on the exponential function with base \( e \), specifically \( e^x \). This is a fundamental part of calculus due to its unique properties, such as its derivative and integral being proportional to the function itself.
The function \( e^x \) grows faster than any polynomial as \( x \) increases and approaches zero as \( x \) becomes more negative. In the exercise's equation, \( e^{-x} = e^3 \), we identify a scenario where both sides are exponential functions with the same base. This relationship allows us to equate the exponents directly, hence simplifying our equation solving process. Exponential functions often appear in natural phenomena like population growth, radioactive decay, and more, illustrating their power and versatility in modeling real-world situations.
The function \( e^x \) grows faster than any polynomial as \( x \) increases and approaches zero as \( x \) becomes more negative. In the exercise's equation, \( e^{-x} = e^3 \), we identify a scenario where both sides are exponential functions with the same base. This relationship allows us to equate the exponents directly, hence simplifying our equation solving process. Exponential functions often appear in natural phenomena like population growth, radioactive decay, and more, illustrating their power and versatility in modeling real-world situations.
Equation Solving
Solving equations involves finding the value of the variable that makes the equation true. The given exercise challenges us to manipulate and simplify logarithmic and exponential expressions to find the solution.
In the provided steps, we use properties of logs and exponents to convert a complicated nested logarithmic equation into a simple linear form. This process typically involves:
In the provided steps, we use properties of logs and exponents to convert a complicated nested logarithmic equation into a simple linear form. This process typically involves:
- Rewriting the equation using logarithmic properties
- Isolating the variable
- Applying inverse operations to simplify
Analytic Methods
Analytic methods in mathematics refer to techniques that derive exact solutions using algebraic manipulations rather than approximations or graphical interpretations. For logarithmic equations like \( \ln(\ln e^{-x}) = \ln 3 \), analytic methods prove invaluable, providing precise solutions with logical steps.
These techniques involve properties and rules of logarithms and exponentials that we systematically apply to solve equations. For our exercise, the steps are laid out clearly: from equating logarithms to eventually solving for \( x \) by isolating it analytically. By relying on properties such as \( \ln a = \ln b \Rightarrow a = b \) and knowing how to exponentiate ln expressions to eliminate them, students can pinpoint the exact solution: \( x = -3 \). Unlike numerical methods or graphical solutions, analytic methods give exact, non-approximated answers, which is crucial for rigorous mathematical proof and understanding. By practicing these methods, students enhance their ability to solve complex equations confidently and accurately.
These techniques involve properties and rules of logarithms and exponentials that we systematically apply to solve equations. For our exercise, the steps are laid out clearly: from equating logarithms to eventually solving for \( x \) by isolating it analytically. By relying on properties such as \( \ln a = \ln b \Rightarrow a = b \) and knowing how to exponentiate ln expressions to eliminate them, students can pinpoint the exact solution: \( x = -3 \). Unlike numerical methods or graphical solutions, analytic methods give exact, non-approximated answers, which is crucial for rigorous mathematical proof and understanding. By practicing these methods, students enhance their ability to solve complex equations confidently and accurately.
Other exercises in this chapter
Problem 112
Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Che
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Use any method (analytic or graphical) to solve each equation. $$\log x^{2}=(\log x)^{2}$$
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Use any method (analytic or graphical) to solve each equation. $$e^{x+\ln 3}=4 e^{x}$$
View solution Problem 116
Life Span of Robins \(\quad\) Use the equation $$y=\frac{2-\log (100-x)}{0.42}$$ from Example 11 for Exercises 115 and 116. Estimate the number of years elapsed
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