/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 Solve each equation. $$ 3^{2... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 71

Solve each equation. $$ 3^{2 x-5}=9 $$

Short Answer

Expert verified
x = 3.5

Step by step solution

01

- Understand the equation

The given equation is \(3^{2x-5} = 9\). Notice that 9 can be written as a power of 3.
02

- Rewrite 9 as a power of 3

Recognize that \(9 = 3^2\). Thus, we can rewrite the equation as \(3^{2x-5} = 3^2\).
03

- Equate the exponents

Since the bases are the same, we can set the exponents equal to each other: \(2x - 5 = 2\).
04

- Solve for x

Isolate \(x\) by first adding 5 to both sides: \(2x - 5 + 5 = 2 + 5\), which simplifies to \(2x = 7\). Then, divide both sides by 2: \(x = \frac{7}{2}\) or \(x = 3.5\).

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Key Concepts

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

solving exponential equations
To solve exponential equations, you need to follow a systematic approach. An exponential equation is one in which the variable appears in the exponent. For example, consider the equation \(3^{2x-5} = 9\). Here, the variable \(x\) is in the exponent. Solving such equations involves several key steps, which we'll break down in the following sections.
rewriting bases
A critical step in solving exponential equations is rewriting expressions with the same base. In our example, \(3^{2x-5} = 9\), notice that 9 can be expressed as a power of 3. We can write 9 as \(3^2\). Rewriting the equation this way allows us to work with the same base: \(3^{2x-5} = 3^2\). This simplification is essential because it allows us to equate the exponents.
equating exponents
Once the bases are the same, as in \(3^{2x-5} = 3^2\), we can set the exponents equal to each other. This is because if \(a^m = a^n\), then \(m = n\). In our equation, we can now write: \(2x - 5 = 2\). By equating the exponents, we turn our exponential equation into a simpler linear equation that is easier to solve.
isolating variables
The final step is to isolate the variable. From \(2x - 5 = 2\), we solve for \(x\) by first adding 5 to both sides: \(2x = 7\). Next, we divide by 2: \(x = 3.5\). Always remember to follow basic algebraic principles to isolate the variable, ensuring to perform the same operations on both sides of the equation to maintain equality.

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