Math Problem Statement

Solve for x in the given set of exponential equations.

Solution

Let's go through each equation and solve for xx.

1) 162x3=4x+216^{2x - 3} = 4^{x + 2}

Express both sides with base 2:

16=24and4=2216 = 2^4 \quad \text{and} \quad 4 = 2^2

Thus, the equation becomes:

(24)2x3=(22)x+2(2^4)^{2x - 3} = (2^2)^{x + 2}

Simplify:

24(2x3)=22(x+2)2^{4(2x - 3)} = 2^{2(x + 2)}

Equating exponents:

4(2x3)=2(x+2)4(2x - 3) = 2(x + 2)

Solve for xx:

8x12=2x+48x - 12 = 2x + 4

6x=166x = 16

x=83x = \frac{8}{3}


2) (12)2x=23x\left(\frac{1}{2}\right)^{2x} = 2^{3 - x}

Rewrite 12\frac{1}{2} as 212^{-1}:

(21)2x=23x(2^{-1})^{2x} = 2^{3 - x}

Simplify:

22x=23x2^{-2x} = 2^{3 - x}

Equating exponents:

2x=3x-2x = 3 - x

Solve for xx:

2x+x=3-2x + x = 3

x=3x=3-x = 3 \quad \Rightarrow \quad x = -3


3) 42x+7322x34^{2x+7} \leq 32^{2x-3}

Express both sides with base 2:

4=22and32=254 = 2^2 \quad \text{and} \quad 32 = 2^5

Rewrite the equation:

(22)2x+7(25)2x3(2^2)^{2x + 7} \leq (2^5)^{2x - 3}

Simplify:

22(2x+7)25(2x3)2^{2(2x + 7)} \leq 2^{5(2x - 3)}

Equating exponents:

4x+1410x154x + 14 \leq 10x - 15

Solve for xx:

14+1510x4x14 + 15 \leq 10x - 4x

296x29 \leq 6x

x296x \geq \frac{29}{6}


4) (25)5x1254\left(\frac{2}{5}\right)^{5x - 1} \geq \frac{25}{4}

Rewrite 254\frac{25}{4} as (52)2\left(\frac{5}{2}\right)^2:

(25)5x1(52)2\left(\frac{2}{5}\right)^{5x - 1} \geq \left(\frac{5}{2}\right)^2

Taking logarithms (or using exponents properties):

5x125x - 1 \leq -2

Solve for xx:

5x15x \leq -1

x15x \leq -\frac{1}{5}


5) (23)5x+2=(32)2x\left(\frac{2}{3}\right)^{5x + 2} = \left(\frac{3}{2}\right)^{2x}

Take the logarithm of both sides:

(5x+2)log(23)=2xlog(32)(5x + 2) \log\left(\frac{2}{3}\right) = 2x \log\left(\frac{3}{2}\right)

Note that log(23)=log(32)\log\left(\frac{2}{3}\right) = -\log\left(\frac{3}{2}\right). The equation becomes:

(5x+2)=2x-(5x + 2) = 2x

Solve for xx:

5x2=2x-5x - 2 = 2x

7x=2-7x = 2

x=27x = -\frac{2}{7}


Would you like me to continue with the rest of the questions, or do you need further clarification on these solutions? Here are some related questions you might be interested in:

  1. How do logarithmic properties help solve exponential equations?
  2. Can all exponential equations be solved by converting to the same base?
  3. What are the general steps to solving inequalities involving exponents?
  4. How do you handle fractional bases in exponential equations?
  5. Why do exponential functions often appear in growth or decay models?

Tip: When solving equations involving exponents, converting both sides to the same base often simplifies the process significantly!

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Math Problem Analysis

Mathematical Concepts

Exponential Equations
Algebra
Logarithms
Inequalities

Formulas

a^x = b^y → x = y (if a = b)
Logarithmic properties
Properties of exponents

Theorems

Laws of Exponents
Logarithmic identities
Solving exponential equations by equating exponents

Suitable Grade Level

Grades 9-12