Problem 37
Question
Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln (x-1)=2$$
Step-by-Step Solution
Verified Answer
The solution to the equation \(\ln (x-1) = 2\) is \(x = e^2 + 1\).
1Step 1: Isolate the logarithmic expression
This step is quite straightforward as the logarithmic expression \(\ln (x-1)\) is already isolated and equal to 2.
2Step 2: Remove the logarithm
Make use of the property of exponentiation to get rid of the natural logarithm. The property states that if \(\ln a = b\), then \(a = e^b\). So from \(\ln (x-1) = 2\), we get \(x - 1 = e^2\).
3Step 3: Solve for x
Rearrange the equation to solve for \(x\) by adding 1 to both sides. This results in \(x = e^2 + 1\).
4Step 4: Check for extraneous solutions
This step involves substituting the value of \(x\) back into the original equation and checking if both sides are equal. Since natural logarithm is defined for all positive real numbers, and \(e^2 + 1 > 1\), the solution is valid.
Key Concepts
Understanding the Natural LogarithmThe Process of ExponentiationIdentifying Extraneous Solutions
Understanding the Natural Logarithm
The natural logarithm, often denoted as \(\ln\), is a special type of logarithm that has the mathematical constant \(e\) as its base. The constant \(e\), approximately 2.718, is known as Euler's number and is fundamental in mathematics, particularly in calculus and complex analysis.
The natural logarithm is used extensively in scientific fields such as physics, biology, and economics because of its natural occurrence in many growth processes. When you see \(\ln(x)\), it represents the time it takes for a quantity growing exponentially to reach \(x\), assuming the quantity starts at 1 and grows at a rate of \(e^t\).
Key properties of the natural logarithm include:
The natural logarithm is used extensively in scientific fields such as physics, biology, and economics because of its natural occurrence in many growth processes. When you see \(\ln(x)\), it represents the time it takes for a quantity growing exponentially to reach \(x\), assuming the quantity starts at 1 and grows at a rate of \(e^t\).
Key properties of the natural logarithm include:
- \(\ln(1) = 0\) because \(e^0 = 1\).
- The function is only defined for \(x > 0\), which means the argument of the natural logarithm must be positive.
- The natural logarithm is the inverse operation of exponentiation with base \(e\).
The Process of Exponentiation
Exponentiation is one of the fundamental operations in mathematics, involving two numbers: the base and the exponent. In the expression \(a^b\), \(a\) is raised to the power of \(b\).
Exponentiation with base \(e\) plays a significant role when solving equations involving natural logarithms. When you have an equation like \(\ln(x-1) = 2\), exponentiation can be used to "eliminate" the natural logarithm. This is done using the property: if \(\ln a = b\), then \(a = e^b\). In this context, the operation translates the equation into an exponential equation: \(x-1 = e^2\).
Exponentiation with base \(e\) plays a significant role when solving equations involving natural logarithms. When you have an equation like \(\ln(x-1) = 2\), exponentiation can be used to "eliminate" the natural logarithm. This is done using the property: if \(\ln a = b\), then \(a = e^b\). In this context, the operation translates the equation into an exponential equation: \(x-1 = e^2\).
- Exponentiation is a way of expressing repeated multiplication. For example, \(3^4\) is the same as \(3 \times 3 \times 3 \times 3\).
- It is important to remember that any number raised to the power of 0 is 1 (e.g., \(e^0 = 1\)).
- Understanding how to work backwards from a logarithm through exponentiation is a crucial skill in algebra and calculus.
Identifying Extraneous Solutions
In mathematics, especially when dealing with logarithmic and radical equations, extraneous solutions are solutions that arise from the algebraic process but do not satisfy the original equation.
After solving an equation, it is vital to check whether the solutions are valid within the context of the original problem. When solving \(\ln(x-1) = 2\), even though the calculated solution \(x = e^2 + 1\) looks correct algebraically, it is crucial to verify by substituting back into the original equation.
Key points about extraneous solutions:
After solving an equation, it is vital to check whether the solutions are valid within the context of the original problem. When solving \(\ln(x-1) = 2\), even though the calculated solution \(x = e^2 + 1\) looks correct algebraically, it is crucial to verify by substituting back into the original equation.
Key points about extraneous solutions:
- They can emerge when both sides of an equation are squared or when logarithms are isolated.
- Always substitute the solution back into the original equation to ensure that it does not produce a mathematical error (like taking a logarithm of a non-positive number).
- In our exercise, since \(e^2 + 1 > 1\), the value of \(x\) is valid and no extraneous solutions exist.
Other exercises in this chapter
Problem 37
Evaluate the expression to four decimal places using a calculator. $$\ln \sqrt{2}$$
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Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-4 x+\frac{1}{5}$$
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The population of white-tailed deer in a wildlife refuge \(t\) months after their introduction into the refuge can be modeled by the logistic function $$N(t)=\f
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In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined
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