Question

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

Short answer

x = 3.5

Step-by-step solution

- Understand the equation

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

- Rewrite 9 as a power of 3

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

- Equate the exponents

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

- 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 \(x = 3.5\).

Key concepts

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.