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Chapter 4: Problem 23

Solve the following equations for \(x\) \(3(2.7)^{5 x}=8.1\)

Short Answer

Expert verified
x = \frac{1}{5}

Step by step solution

01

- Divide both sides by 3

Start by isolating the exponential expression on one side of the equation. Divide both sides by 3:equation: \( \frac{3(2.7)^{5 x}}{3} = \frac{8.1}{3} \)which simplifies to:\( (2.7)^{5 x} = 2.7 \)
02

- Recognize exponential base

Notice that 2.7 in the right hand side of the equation \( (2.7)^{5 x} = 2.7 \) can be rewritten as \( (2.7)^1 \) or simply 2.7.
03

- Equate the exponents

Since the bases on both sides of the equation are equal, equate the exponents:\( 5x = 1 \)
04

- Solve for x

Divide both sides by 5 to isolate x:\( x = \frac{1}{5} \)

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Key Concepts

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

Isolate the Exponential Expression
When solving exponential equations, the first step is to isolate the exponential expression. This makes handling the equation much more straightforward. For the equation \[ 3(2.7)^{5x} = 8.1 \], we can see the exponential part of the equation is \( (2.7)^{5x} \).
To isolate \( (2.7)^{5x} \), we divide both sides of the equation by 3. Doing this simplifies the equation to: \[ (2.7)^{5x} = 2.7 \].
Isolating the exponential expression helps in making the equation easier to manage and solve.
Exponential Base
Understanding the exponential base is crucial when solving exponential equations. The exponential base is the number that is raised to a power. In the equation \[ (2.7)^{5x} = 2.7 \], the base is 2.7.
On both sides of the equation, the base remains the same. Recognizing that \( 2.7 \) on the right-hand side can be written as \( (2.7)^1 \) simplifies our next steps.
This common base approach is integral because it allows us to directly compare the exponents in the next step.
Equate the Exponents
With an isolated exponential expression and a common base, the next step is to equate the exponents. Since we have the equation \[ (2.7)^{5x} = (2.7)^1 \], and the bases are equal, it implies \( 5x = 1 \).
This principle comes from the property of exponents: if \( a^m = a^n \), then \( m = n \). By matching the exponents, we simplify the equation significantly, making it a matter of basic algebra to solve for \( x \).
Solve for x
Finally, to solve for \( x \), we need to isolate it. From the equation \( 5x = 1 \), we solve for \( x \) by dividing both sides by 5. This results in: \[ x = \frac{1}{5} \].
Solving for \( x \) often involves these simpler algebraic steps once the exponents have been equated.
It鈥檚 important to recheck our steps to ensure accuracy. In our case, the isolated variable is clearly and accurately determined as \( x = \frac{1}{5} \).
By following these methodical steps - isolate, recognize the base, equate, and solve - complex exponential equations become manageable and straightforward.

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