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Chapter 12: Problem 25

Solve each equation for \(x .\) See Example 4. $$32^{2 x-3}=2$$

Short Answer

Expert verified
\(x = \frac{8}{5}\).

Step by step solution

01

Rewrite the Base Using the Same Base

Both 32 and 2 can be expressed as powers of 2. Note that \(32 = 2^5\). Therefore, the equation \(32^{2x-3} = 2\) can be rewritten as \((2^5)^{2x-3} = 2^1\).
02

Apply the Power of a Power Property

Using the property \((a^m)^n = a^{m \cdot n}\), rewrite the left side of the equation: \((2^5)^{2x-3} = 2^{5(2x-3)} = 2^{10x-15}\). So now the equation is \(2^{10x-15} = 2^1\).
03

Set Exponents Equal

Since the bases are the same (both are power of 2), you can set the exponents equal to each other: \(10x - 15 = 1\).
04

Solve for x

Solve the equation \(10x - 15 = 1\) by first adding 15 to both sides, giving \(10x = 16\). Then, divide both sides by 10 to find \(x = \frac{16}{10}\), which simplifies to \(x = \frac{8}{5}\).

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Key Concepts

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

Power of a Power Property
Understanding the "power of a power" property is crucial when solving exponential equations. It simplifies expressions where an exponent is raised to another power, like \((a^m)^n\). The property formula is:
  • \((a^m)^n = a^{m \cdot n}\).
Using this rule allows you to change complex exponential expressions into simpler forms. In the exercise, we start with \((2^5)^{2x-3}\).
This is transformed into \(2^{5 \cdot (2x-3)}\), which simplifies to \(2^{10x - 15}\). The key takeaway is to multiply the exponents together. This move also helps in balancing both sides of the equation by ensuring that the base only needs focus.
Once you simplify the powers, comparing the exponents becomes straightforward.
Setting Exponents Equal
Once the exponential expression is simplified, the next step is to set the exponents equal. This is only possible if the bases on both sides of the equation are the same.
In our example, both sides have been rewritten to base 2, resulting in the equation:
  • \(2^{10x - 15} = 2^1\).
This allows us to disregard the base 2 (since they're equal) and just equate the exponents:
  • \(10x - 15 = 1\).
By setting the exponents equal, we transform a potentially complex exponential equation into a simple linear equation.
This greatly eases the process of finding the value of \(x\). Remember, equating exponents is only valid when the bases are the same, so always check this before proceeding.
Solving for x
After setting the exponents equal, the final step is solving the resulting equation for \(x\). The equation from our previous steps is:
  • \(10x - 15 = 1\).
To solve, follow these simple algebraic steps:
  • Add 15 to both sides to isolate terms with \(x\):
    \(10x = 16\).
  • Divide each side by 10 to solve for \(x\):
    \(x = \frac{16}{10}\).
This will simplify to \(x = \frac{8}{5}\). By breaking it down to simple arithmetic processes, solving for \(x\) becomes a manageable task. Each step in the process is about applying basic algebra rules to isolate \(x\) effectively, allowing you to find the precise solution.

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Most popular questions from this chapter

Solve each equation for \(x .\) See Example 4. $$81^{x-1}=27^{2 x}$$

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