Problem 93
Question
This set of exercises will draw on the ideas presented in this section and your general math background. What is wrong with the following step? $$2^{x+5}=3^{4 x} \Rightarrow x+5=4 x$$
Step-by-Step Solution
Verified Answer
The mistake in the step comes from wrongly assuming that exponents with different bases can be equated. The equation \(2^{x+5} = 3^{4x}\) does not imply \(x+5 = 4x\), since the bases (2 and 3) are different. Thus, the statement is incorrect.
1Step 1: Identify the Error
The step equates exponents of different bases (2 and 3). This is only valid when bases are equal.
2Step 2: Correct Approach
Take logarithms: \((x+5)\ln 2 = 4x\ln 3\), then solve for \(x\).
Key Concepts
Faulty Operations in AlgebraExponents and BasesPrecalculus Problem Solving
Faulty Operations in Algebra
One common pitfall in algebra is performing faulty operations without a proper understanding of the mathematical rules. This often occurs when students try to simplify complex expressions or solve equations too quickly without due attention to detail.
For example, in the operation \(2^{x+5} = 3^{4x} \Rightarrow x+5 = 4x\), there is an incorrect assumption that since the exponents on both sides look somewhat comparable, they might be directly set equal to each other. This is a classic example of a faulty operation in algebra. Operations on expressions with exponents require a firm grasp of the rules that govern their manipulation.
In dealing with exponentials, it's vital to remember that you can only equate the bases and exponents separately if the bases are the same. If the bases are different, we cannot equate the exponents directly, as each base has its unique properties and growth rates. Instead, other strategies must be employed, such as taking the logarithm of both sides or using other algebraic techniques to solve for the variable.
For example, in the operation \(2^{x+5} = 3^{4x} \Rightarrow x+5 = 4x\), there is an incorrect assumption that since the exponents on both sides look somewhat comparable, they might be directly set equal to each other. This is a classic example of a faulty operation in algebra. Operations on expressions with exponents require a firm grasp of the rules that govern their manipulation.
In dealing with exponentials, it's vital to remember that you can only equate the bases and exponents separately if the bases are the same. If the bases are different, we cannot equate the exponents directly, as each base has its unique properties and growth rates. Instead, other strategies must be employed, such as taking the logarithm of both sides or using other algebraic techniques to solve for the variable.
Exponents and Bases
Understanding exponents and bases is central to working with exponential equations. An exponential expression consists of a base raised to a power, represented as \(b^n\), where \(b\) is the base and \(n\) is the exponent.
The base of an exponent tells us what number is being multiplied by itself, and the exponent indicates how many times this multiplication occurs. It's important to understand that different bases will grow at different rates. For instance, \(2^3\) is 8, but \(3^2\) is 9, although the numbers 2 and 3 are close, their exponential forms are not equal due to the nature of their bases.
When solving equations involving exponents, such as \(2^{x+5} = 3^{4x}\), we cannot simply equate the powers because the bases - 2 and 3 - are different. Instead, we would typically employ logarithms to solve for \(x\) because logarithms allow us to 'unpack' the exponent and bring it down to a place where we can handle it algebraically. This necessary understanding of how exponents work with different bases is often the key to finding the right solution method for exponential equations.
The base of an exponent tells us what number is being multiplied by itself, and the exponent indicates how many times this multiplication occurs. It's important to understand that different bases will grow at different rates. For instance, \(2^3\) is 8, but \(3^2\) is 9, although the numbers 2 and 3 are close, their exponential forms are not equal due to the nature of their bases.
When solving equations involving exponents, such as \(2^{x+5} = 3^{4x}\), we cannot simply equate the powers because the bases - 2 and 3 - are different. Instead, we would typically employ logarithms to solve for \(x\) because logarithms allow us to 'unpack' the exponent and bring it down to a place where we can handle it algebraically. This necessary understanding of how exponents work with different bases is often the key to finding the right solution method for exponential equations.
Precalculus Problem Solving
Precalculus is a course that includes various topics necessary for understanding calculus. Problem solving in precalculus often involves dealing with exponential functions, trigonometry, and algebraic functions. Key to success in this subject is mastering a wide range of techniques and having the flexibility to determine which method is most suitable for the problem at hand.
Let's consider solving the exponential equation \(2^{x+5} = 3^{4x}\). A common strategy for solving such exponential equations when the bases are different is to employ logarithms. By applying logarithms to both sides of the equation, we can bring down the exponents and solve for \(x\) algebraically. It is important to be patient and methodical when choosing a problem-solving strategy to avoid common mistakes like equating exponents with different bases.
As students build their problem-solving toolbox in precalculus, they should practice identifying the nature of the problem and select a strategy based on a solid understanding of the concepts involved. Knowing when to apply logarithms, when to factor, or when to graph a function, forms the foundation for approaching a wide array of precalculus problems effectively.
Let's consider solving the exponential equation \(2^{x+5} = 3^{4x}\). A common strategy for solving such exponential equations when the bases are different is to employ logarithms. By applying logarithms to both sides of the equation, we can bring down the exponents and solve for \(x\) algebraically. It is important to be patient and methodical when choosing a problem-solving strategy to avoid common mistakes like equating exponents with different bases.
As students build their problem-solving toolbox in precalculus, they should practice identifying the nature of the problem and select a strategy based on a solid understanding of the concepts involved. Knowing when to apply logarithms, when to factor, or when to graph a function, forms the foundation for approaching a wide array of precalculus problems effectively.
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