Chapter 18: Problem 25
Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\ln (5 x+2)-\ln (x+6)=\ln 4$$
Short Answer
Expert verified
The value of x that satisfies the equation is \(x = 22\).
Step by step solution
01
Apply the Logarithmic Property
Use the property of logarithms that allows the subtraction of two logs with the same base to be rewritten as the log of a quotient: \[ \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \] By applying this property, the equation becomes:\[ \ln\left(\frac{5x + 2}{x + 6}\right) = \ln(4) \]
02
Equate the Arguments of the Logarithms
Since the logarithms are equal and have the same base, their arguments must also be equal. Set the arguments equal to each other: \[ \frac{5x + 2}{x + 6} = 4 \]
03
Solve for x
Clear the fraction by multiplying both sides of the equation by \(x + 6\): \[ (5x + 2) = 4(x + 6) \] Then expand and simplify: \[ 5x + 2 = 4x + 24 \] \[ 5x - 4x = 24 - 2 \] \[ x = 22 \]
04
Check the Solution
Substitute the value of x back into the original equation to verify: \[ \ln(5(22) + 2) - \ln(22 + 6) = \ln 4 \] Simplify: \[ \ln(112) - \ln(28) = \ln 4 \] Since \ln(112) - \ln(28) is indeed \ln 4, the solution \(x = 22\) is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Properties
Understanding logarithmic properties is essential when solving logarithmic equations. A logarithm, by definition, is the power to which a number must be raised to obtain some other number. For example, if we have the base 'e' (where 'e' is the mathematical constant approximately equal to 2.71828), represented as \(\ln(x)\), we are asking 'to what power must e be raised to produce x?'.
The logarithmic property used in our exercise is known as the quotient rule. It states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator, as long as the logs have the same base. This is formally written as \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\). Applying this property can dramatically simplify the process of solving logarithmic equations as it allows us to combine two separate logarithmic expressions into one.
Another critical property is the equality of logarithms. If \(\ln(a) = \ln(b)\), then \({a = b}\) since the logarithmic function is one-to-one. This principle is key in our step-by-step solution where the equation \(\ln\left(\frac{5x + 2}{x + 6}\right) = \ln(4)\) implies that \(\frac{5x + 2}{x + 6} = 4\).
The logarithmic property used in our exercise is known as the quotient rule. It states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator, as long as the logs have the same base. This is formally written as \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\). Applying this property can dramatically simplify the process of solving logarithmic equations as it allows us to combine two separate logarithmic expressions into one.
Another critical property is the equality of logarithms. If \(\ln(a) = \ln(b)\), then \({a = b}\) since the logarithmic function is one-to-one. This principle is key in our step-by-step solution where the equation \(\ln\left(\frac{5x + 2}{x + 6}\right) = \ln(4)\) implies that \(\frac{5x + 2}{x + 6} = 4\).
Logarithm Rules
Logarithm rules, also known as log rules, are a set of guidelines that allow us to manipulate logarithmic expressions systematically. The quotient rule utilized in the given problem is one of these essential log rules. In addition to the quotient rule, there are other important rules such as the product rule and the power rule.
The product rule tells us that the log of a product is the sum of the logs: \(\ln(ab) = \ln(a) + \ln(b)\). The power rule states that the log of a power is the exponent times the log of the base: \(\ln(a^b) = b \ln(a)\). These rules make it possible to break down complex logarithmic expressions into simpler forms that are easier to solve and understand.
Mastering these log rules is crucial because it allows students to approach logarithmic equations with a toolkit of simplification strategies, making them less intimidating and more manageable. Recognizing which rule to apply and when also improves students’ problem-solving skills.
The product rule tells us that the log of a product is the sum of the logs: \(\ln(ab) = \ln(a) + \ln(b)\). The power rule states that the log of a power is the exponent times the log of the base: \(\ln(a^b) = b \ln(a)\). These rules make it possible to break down complex logarithmic expressions into simpler forms that are easier to solve and understand.
Mastering these log rules is crucial because it allows students to approach logarithmic equations with a toolkit of simplification strategies, making them less intimidating and more manageable. Recognizing which rule to apply and when also improves students’ problem-solving skills.
Exponential Functions
Exponential functions are closely related to logarithms, as they can be considered inverse operations. An exponential function is of the form \({f(x) = a^x}\), where 'a' is a positive constant, and 'x' is the exponent. The function \({e^x}\), where 'e' is Euler's number, is a special type of exponential function known as the natural exponential function.
In the context of our problem, understanding exponential functions helps us comprehend what a logarithmic function represents. The equation \({e^{\ln(x)} = x}\) demonstrates this inverseness. If \({x = e^y}\), then \(\ln(x) = y\). This relationship is pivotal when solving logarithmic equations because it allows us to switch between exponential and logarithmic forms, whichever is more conducive to finding a solution.
Additionally, the properties of exponential functions, such as continuous growth or decay, are used to describe various real-world phenomena. Therefore, mastering both logarithmic and exponential functions provides a strong mathematical foundation for understanding complex concepts in calculus, physics, finance, and many other fields.
In the context of our problem, understanding exponential functions helps us comprehend what a logarithmic function represents. The equation \({e^{\ln(x)} = x}\) demonstrates this inverseness. If \({x = e^y}\), then \(\ln(x) = y\). This relationship is pivotal when solving logarithmic equations because it allows us to switch between exponential and logarithmic forms, whichever is more conducive to finding a solution.
Additionally, the properties of exponential functions, such as continuous growth or decay, are used to describe various real-world phenomena. Therefore, mastering both logarithmic and exponential functions provides a strong mathematical foundation for understanding complex concepts in calculus, physics, finance, and many other fields.
