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Chapter 5: Problem 46

Solve the equation. $$\ln (3 x+5)-1=\ln (2 x-3)$$

Short Answer

Expert verified
**Question:** Solve the given equation or determine if there is no solution within the domain of the equation: $$\ln(3x + 5) = \ln(2x - 3)$$ **Answer:** There is no solution within the domain of the given equation.

Step by step solution

01

Use the properties of logarithm to simplify the equation

Since the logarithms on both sides have the same base, we can set the argument of the logarithms equal: $$3x + 5 - 1 = 2x - 3$$
02

Solve the resulting linear equation

First, simplify the equation and move all the x terms to one side: $$3x + 4 = 2x - 3$$ $$3x - 2x = -3 - 4$$ $$x = -7$$
03

Check the solution within the domain of the initial equations

Now we need to check if the solution is valid by plugging in the value of x into the initial logarithm arguments. The arguments should be greater than 0, since ln(x) is undefined for nonpositive x values. For the first argument: $$3(-7) + 5 = -16$$ For the second argument: $$2(-7) - 3 = -17$$ Both arguments are not greater than 0, meaning that x = -7 is not within the domain of the initial equations. Therefore, our final answer is: There is no solution within the domain of the given equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Logarithms
Logarithms are mathematical tools used to solve equations, especially when dealing with exponential growth or decay. One critical property is that if you have logarithms with the same base on both sides of an equation, their arguments can be set equal to each other. This is because the logarithmic function is one-to-one, meaning if \( \ln(a) = \ln(b) \), then \( a = b \). This property is very useful in solving equations that involve logarithms because it simplifies the equation significantly.
In the exercise, the logarithmic terms \( \ln(3x + 5) \) and \( \ln(2x - 3) \) are given with the same base of \( e \). As such, we equate their arguments directly: \( 3x + 5 = 2x - 3 \), excluding the constant from the left side for simplicity. This allowed us to solve for \( x \) using basic linear equation techniques, showing the power of logarithm properties in simplifying our work.
Solving Linear Equations
Linear equations, such as the one derived from the logarithmic equation, are straightforward to solve.
The goal is to isolate the variable (in this case \( x \)) on one side of the equation. You achieve this by strategically adding, subtracting, multiplying, or dividing both sides of the equation.
The transformed equation \( 3x + 4 = 2x - 3 \) was simplified further by subtracting \( 2x \) from both sides, resulting in \( x + 4 = -3 \).
This equation was solved by subtracting 4 from both sides, leading to the solution \( x = -7 \).
  • Ensure all like terms are combined.
  • Perform inverse operations to get the variable on one side.
  • Check for shortcuts like eliminating terms using addition or subtraction.
These steps are fundamental when tackling linear equations.
Domain of a Function
The domain of a function refers to all possible input values for which the function is defined. In the context of logarithms, it's crucial because logarithmic functions are only defined for positive arguments.
For example, the function \( \ln(x) \) is undefined for \( x \leq 0 \). Therefore, every argument inside a logarithm must be greater than zero.
In our original equation with \( \ln(3x + 5) \) and \( \ln(2x - 3) \), the arguments \( 3x + 5 \) and \( 2x - 3 \) should both be positive for the logarithm to exist.
If we substitute \( x = -7 \) from the solution back into these expressions:
  • \( 3(-7) + 5 = -16 \)
  • \( 2(-7) - 3 = -17 \)
Both values are negative, indicating \( x = -7 \) does not lie within the domain.
This highlights the importance of verifying solutions within the functional domain, ensuring solutions exist and are mathematically valid.

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Most popular questions from this chapter

In the past two decades, more women than men have been entering college. The table shows the percentage of male first-year college students in selected years. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { Year } & 1985 & 1990 & 1995 & 1997 & 1998 & 1999 & 2003 & 2004 & 2005 \\ \hline \text { Percent } & 48.9 & 46.9 & 45.6 & 45.5 & 45.5 & 45.3 & 45.1 & 44.9 & 45.0 \\ \hline \end{array}$$ (a) Find three models for this data: exponential, logarithmic, and power, with \(x=5\) corresponding to 1985 (b) For the years \(1985-2005,\) is there any significant difference among the models? (c) Assume that the models remain accurate. What year does each predict as the first year in which fewer than \(43 \%\) of first-year college students will be male? (d) We actually have some additional data: $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2000 & 2001 & 2002 & 2006 \\ \hline \text { Percent } & 45.2 & 44.9 & 45.0 & 45.1 \\ \hline \end{array}$$ Which model did the best job of predicting the new data?

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). Under normal conditions, the atmospheric pressure (in millibars) at height \(h\) feet above sea level is given by \(P(h)=\) \(1015 e^{-k t},\) where \(k\) is a positive constant. (a) If the pressure at 18,000 feet is half the pressure at sea level, find \(k\). (b) Using the information from part (a), find the atmospheric pressure at 1000 feet, 5000 feet, and 15,000 feet.

Sketch a complete graph of the function. $$f(x)=3^{-x}$$

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The amount \(P\) of ozone in the atmosphere is currently decaying exponentially each year at a continuous rate of \(\frac{1}{4} \%\) (that is, \(k=-.0025\) ). How long will it take for half the ozone to disappear (that is, when will the amount be \(P / 2\) )? [Your answer is the half-life of ozone.]

The U.S. Department of Commerce estimated that there were 54 million Internet users in the United States in 1999 and 85 million in 2002 . (a) Find an exponential function that models the number of Internet users in year \(x,\) with \(x=0\) corresponding to 1999 (b) For how long is this model likely to remain accurate? [Hint: The current U.S. population is about 305 million.]
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