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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
The solution for x is \( \frac{1}{5} \).

Step by step solution

01

Isolate the Exponential Expression

Divide both sides of the equation by 3 to isolate the exponential expression: \[ (2.7)^{5x} = \frac{8.1}{3}\]
02

Simplify the Division

Simplify the right side of the equation: \[ (2.7)^{5x} = 2.7 \]
03

Set Exponents Equal

Since the bases are the same, set the exponents equal to each other: \[ 5x = 1 \]
04

Solve for x

Divide both sides by 5 to solve for \(x\): \[ x = \frac{1}{5} \]

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Key Concepts

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

isolate exponential expression
To solve exponential equations, the first step is often to isolate the exponential expression. This means we want to get the term with the exponent by itself on one side of the equation.

In the given exercise, we start with the equation: 3(2.7)^{5x} = 8.1.

We need to isolate (2.7)^{5x}.

To do this, we divide both sides of the equation by 3:

\( (2.7)^{5x} = \frac{8.1}{3} \)

This leaves us with (2.7)^{5x} on one side, which is exactly what we want.
simplify division
After isolating the exponential expression, our next task is simplifying the division.

In the current scenario, the right-hand side of the equation is \( \frac{8.1}{3} \).

We perform the division: 8.1 divided by 3.

This gives us 2.7:

\( (2.7)^{5x} = 2.7 \).

Here, we see that the result of the division simplifies our equation significantly.

Now, both sides of the equation share the same base of 2.7, which brings us closer to solving for the variable.
set exponents equal
When both sides of the equation have the same base, we can set the exponents equal to each other. This is because if \(a^m = a^n\), then \(m = n\).

In this example, both sides have the base 2.7:

\( (2.7)^{5x} = 2.7 \)

We recognize that 2.7 can be written as (2.7)^1.

Therefore, the equation becomes: (2.7)^{5x} = (2.7)^1.

Since the bases are identical, we can equate their exponents:

5x = 1.
solve for variable
The final step is to solve for the variable. We have the equation 5x = 1.

To find the value of \(x\), we divide both sides by 5:

\( x = \frac{1}{5} \).

This simplifies to \( x = 0.2 \).

Therefore, the solution to the equation 3(2.7)^{5x} = 8.1 is \( x = 0.2 \).

Always remember to follow these steps when solving exponential equations, and you will find the solution easily.

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