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Magazine Chapter 4: Problem 36
Solve the equation. $$\ln x=1+\ln (x+1)$$
Short Answer
Expert verified
The solution is approximately \(x \approx -3.986\).
Step by step solution
01
Rewrite the Equation
Start with the equation \(\ln x = 1 + \ln (x + 1)\). Use the property of logarithms that \(a = \ln b\) can be rewritten as \(b = e^a\). This gives us \(x = e^{1 + \ln (x + 1)}\).
02
Apply Properties of Exponents
Apply the exponent rule \(b^{m+n} = b^m \times b^n\). Therefore, \(e^{1 + \ln (x + 1)} = e^1 \times e^{\ln (x + 1)} = e \times (x+1)\).
03
Set Equation and Simplify
The equation \(x = e \times (x + 1)\) simplifies to \(x = e(x + 1)\). Distribute \(e\) to get \(x = ex + e\).
04
Rearrange Terms
Rearrange the terms to isolate \(x\) on one side: \(x - ex = e\).
05
Factor Out x
Factor \(x\) from \(x - ex = e\) to give \(x(1 - e) = e\).
06
Solve for x
Divide both sides by \((1 - e)\) to solve for \(x\): \(x = \frac{e}{1 - e}\).
07
Approximate to Exact Decimal
Calculate \(e \approx 2.718\) and use it to find an approximate numerical answer for \(x = \frac{2.718}{1 - 2.718}\). Thus, \(x \approx -3.986\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
Exponential functions are a type of mathematical function that have the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the variable. They are used to model scenarios where growth or decay occurs at a constant rate.
Exponential functions have several distinct properties:
Exponential functions have several distinct properties:
- They pass through the point \((0, 1)\) when the base, \( a \), is greater than 0.
- If \( 0 < a < 1 \), the function represents decay, while if \( a > 1 \), it represents growth.
- Exponential functions are defined for all real numbers.
- The function \( f(x) = e^x \) is a particularly important example, where \( e \) is the base of the natural logarithm, approximately equal to 2.718.
Properties of Logarithms
Logarithms are the natural inverse of exponential functions. They help to solve for a variable raised to a power. The logarithm of a number is the exponent to which the base must be raised to produce the number. For instance, \( \ln x = y\) means that \( e^y = x \). Here are some essential properties:
- Product Rule: \( \ln(ab) = \ln a + \ln b \)
- Quotient Rule: \( \ln(a/b) = \ln a - \ln b \)
- Power Rule: \( \ln(a^b) = b \cdot \ln a \)
- By definition, \( \ln(e) = 1 \)
Properties of Exponents
The properties of exponents dictate how expressions with powers behave. Exponents give us a shorthand way to express repeated multiplication. Here is a brief overview of these properties, which are pivotal in solving exponential and logarithmic equations:
- Multiplication Rule: \( b^m \times b^n = b^{m+n} \)
- Division Rule: \( \frac{b^m}{b^n} = b^{m-n} \) for \( b eq 0 \)
- Power of a Power Rule: \((b^m)^n = b^{m \times n} \)
- Anything raised to the zero power is 1, i.e., \( b^0 = 1 \)
- The natural base exponent rule states \( e^{\ln(x)} = x \) when \( x > 0 \)
