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Chapter 4: Problem 24

Solve the following equations for \(x\). $$4(2.7)^{2 x-1}=10.8$$

Short Answer

Expert verified
The solution is \(x=1\).

Step by step solution

01

Simplify the equation

Start by dividing both sides of the equation by 4 to isolate the exponential expression. \[4(2.7)^{2x-1}=10.8 \implies (2.7)^{2x-1}=\frac{10.8}{4}\]
02

Simplify the right side

Simplify \(\frac{10.8}{4}\) to get:\[(2.7)^{2x-1}=2.7\]
03

Set exponents equal

Since the bases on both sides are the same, set the exponents equal to each other:\[2x-1=1\]
04

Solve for x

Solve the equation \(2x-1=1\) for \(x\). Add 1 to both sides: \[2x=2\]Then divide both sides by 2: \[x=1\]

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Key Concepts

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

Isolating exponential expressions
When solving exponential equations, one crucial step is to isolate the exponential expression. This means you want the term with the exponent to be by itself on one side of the equation. Let's break it down using the problem given:

The problem is: \(4(2.7)^{2x-1} = 10.8\). To isolate \( (2.7)^{2x-1} \), divide both sides of the equation by 4:
\[ 4(2.7)^{2x-1} = 10.8 \rightarrow (2.7)^{2x-1} = \frac{10.8}{4} \]
This simplification helps to keep our equation neat and easy to manage. Isolating the exponential expression is typically the first move you'd make in solving these types of problems.
Exponents
Exponents are the power to which a number or expression is raised. In this problem, you see \( (2.7)^{2x-1} \). The number 2.7 is the base, and \( 2x-1 \) is the exponent.

We need to understand that any time the bases in the equation are equal, the exponents themselves must also be equal. Here鈥檚 how we simplify it:
After isolating the exponentiated term, we get:
\[ (2.7)^{2x-1} = 2.7 \] This implies that the exponent \( 2x-1 \) must equal 1 because \( 2.7^1 = 2.7 \). So, we set up the equation of the exponents: \[ 2x-1 = 1 \]
Understanding that bases of exponential expressions determine the next steps makes solving much more manageable.
Algebraic Manipulation
Algebraic manipulation involves performing operations to both sides of an equation to isolate the variable we're solving for. In this problem, once we've set the exponents equal, the equation is \(2x-1=1\).

To solve for \(x\):
- Add 1 to both sides to eliminate the constant: \[2x-1+1=1+1 \rightarrow 2x = 2 \]
- Then, divide both sides by 2 to isolate \(x\): \[ \frac{2x}{2} = \frac{2}{2} \rightarrow x = 1 \]
These are straightforward yet essential steps in algebraic manipulation. Clear and methodical processing ensures that we find the correct value of \( x \). Always verify each step to maintain accuracy.

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