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Chapter 4: Problem 27

Solve the following equations for \(x\). $$2^{3 x}=4 \cdot 2^{5 x}$$

Short Answer

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

Step by step solution

01

Simplify the right side of the equation

First, notice that 4 can be rewritten as a power of 2: 4 = 2^2. Thus, the equation becomes: \(2^{3x} = 2^2 \times 2^{5x} \)
02

Use the properties of exponents

Apply the property of exponents that states \(a^m \times a^n = a^{m+n}\). The right side of the equation becomes: \(2^{3x} = 2^{2 + 5x}\)
03

Set the exponents equal to each other

Since the bases are the same, set the exponents equal to each other: \(3x = 2 + 5x\)
04

Solve for x

Rearrange the equation to isolate x. Subtract 3x from both sides: \(0 = 2 + 2x\).Then, subtract 2 from both sides: \(-2 = 2x\).Finally, divide by 2: \(x = -1\)

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Key Concepts

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

Exponent Properties
Understanding exponent properties helps simplify and solve exponential equations such as \(2^{3x} = 4 \cdot 2^{5x}\).
Exponents follow specific rules that make computations easier:
  • For any non-zero number a, \(a^0 = 1\).
  • Multiplication of same base: \((a^m \cdot a^n = a^{m+n})\).
  • Power of a power: \((a^m)^n = a^{m\cdot n}\).
  • Division of same base: \((a^m / a^n = a^{m-n})\).
  • Negative exponents: \(a^{-m} = 1/a^m\).
Using these properties, you can simplify and rewrite exponential expressions. For this example, recognizing that 4 can be expressed as \(2^2\) is key. Therefore, \(4 \cdot 2^{5x}\) becomes \(2^2 \cdot 2^{5x}\). This simplifies using the addition rule for exponents: \(2^{2+5x}\).
Simplifying Expressions
Simplifying exponential expressions is crucial in solving equations. After recognizing \(4 = 2^2\), rewrite the equation:
\(2^{3x} = 2^2 \cdot 2^{5x}\).
Apply the exponent multiplication property where \(a^m \cdot a^n = a^{m+n}\). Here, it becomes \(2^{3x} = 2^{2 + 5x}\).
Simplification leads to easier comparisons. Notice that both sides of the equation now have the same base (2). When bases are identical, compare exponents directly to form a simpler equation: \(3x = 2 + 5x\).
Isolating Variables
Isolating variables means solving for the variable by rearranging the equation. Starting from \(3x = 2 + 5x\):
  • Subtract 3x from both sides: \((0 = 2 + 2x)\).
  • Subtract 2 from both sides: \((-2 = 2x)\).
  • Finally, divide by 2: \((x = -1)\).
By following these steps, we isolate \(x\) and solve the equation. Rearrange equations systematically, ensuring every operation to isolate the variable takes you one step closer to the solution.

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Most popular questions from this chapter

The expressions may be factored as shown. Find the missing factors. $$5^{x+h}+5^{x}=5^{x}(\quad)$$

Which of the following is the same as \(\ln (9 x)-\ln (3 x) ?\) (a) \(\ln 6 x\) (b) \(\ln (9 x) / \ln (3 x)\) (c) \(6 \cdot \ln (x)\) (d) \(\ln 3\)

Calculate values of \(\frac{10^{x}-1}{x}\) for small values of \(x\), and use them to estimate \(\left.\frac{d}{d x}\left(10^{x}\right)\right|_{x=0} .\) What is the formula for \(\frac{d}{d x}\left(10^{x}\right) ?\)

Find \(k\) such that \(2^{x}=e^{k x}\) for all \(x.\)

Find the values of \(x\) at which the function has a possible relative maximum or minimum point. (Recall that \(e^{x}\) is positive for all \(x .\) ) Use the second derivative to determine the nature of the function at these points. $$f(x)=\frac{4 x-1}{e^{x / 2}}$$
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