Math Problem Statement

Solve for x

Solve for x in the given exponential equations: 4^(x+2) = 8^x, 10^(4x+1) > 100^(x-2), 36^(2x+4) = 216^(x+5), 1/81 < 9^(2x-4), 32^(3x-4) > 128^(4x+3).

Solution

Let's go through each of the problems step by step to solve for xx.


6) 4x+2=8x4^{x+2} = 8^x

We start by expressing the bases as powers of 2: 4=22and8=234 = 2^2 \quad \text{and} \quad 8 = 2^3 Rewriting the equation: (22)x+2=(23)x(2^2)^{x+2} = (2^3)^x Simplifying: 22(x+2)=23x2^{2(x+2)} = 2^{3x} This gives: 2x+4=3x2x + 4 = 3x Solving for xx: 4=x4 = x


7) 104x+1>100x210^{4x+1} > 100^{x-2}

Express 100100 as 10210^2: 104x+1>(102)x210^{4x+1} > (10^2)^{x-2} Simplifying: 104x+1>102(x2)10^{4x+1} > 10^{2(x-2)} This gives: 104x+1>102x410^{4x+1} > 10^{2x-4} Now, we can equate the exponents: 4x+1>2x44x + 1 > 2x - 4 Solving for xx: 2x>52x > -5 x>52x > -\frac{5}{2}


8) 362x+4=216x+536^{2x+4} = 216^{x+5}

Express the bases in terms of powers of 6: 36=62and216=6336 = 6^2 \quad \text{and} \quad 216 = 6^3 Rewriting the equation: (62)2x+4=(63)x+5(6^2)^{2x+4} = (6^3)^{x+5} Simplifying: 64x+8=63(x+5)6^{4x+8} = 6^{3(x+5)} Equating the exponents: 4x+8=3x+154x + 8 = 3x + 15 Solving for xx: x=7x = 7


9) 181<92x4\frac{1}{81} < 9^{2x-4}

Express 81 and 9 as powers of 3: 81=34and9=3281 = 3^4 \quad \text{and} \quad 9 = 3^2 Rewriting the equation: 34<(32)2x43^{-4} < (3^2)^{2x-4} Simplifying: 34<34x83^{-4} < 3^{4x-8} Equating the exponents: 4<4x8-4 < 4x - 8 Solving for xx: 4x>44x > 4 x>1x > 1


10) 323x4>1284x+332^{3x-4} > 128^{4x+3}

Express 32 and 128 as powers of 2: 32=25and128=2732 = 2^5 \quad \text{and} \quad 128 = 2^7 Rewriting the equation: (25)3x4>(27)4x+3(2^5)^{3x-4} > (2^7)^{4x+3} Simplifying: 25(3x4)>27(4x+3)2^{5(3x-4)} > 2^{7(4x+3)} Equating the exponents: 15x20>28x+2115x - 20 > 28x + 21 Solving for xx: 13x>41-13x > 41 x<4113x < -\frac{41}{13}


Final Answers:

  1. x=4x = 4
  2. x>52x > -\frac{5}{2}
  3. x=7x = 7
  4. x>1x > 1
  5. x<4113x < -\frac{41}{13}

Let me know if you'd like further details on any of these steps!


Here are 5 additional questions for further exploration:

  1. Can you solve exponential equations using logarithms?
  2. How do you handle inequalities when solving exponential equations?
  3. What happens if the bases of the exponential equations are not powers of the same number?
  4. Can you solve exponential equations with negative exponents?
  5. How would you graph exponential inequalities?

Tip: Always look for common bases in exponential equations to simplify solving!

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Math Problem Analysis

Mathematical Concepts

Exponential equations
Logarithmic inequalities
Properties of exponents
Simplifying expressions

Formulas

a^x = a^y implies x = y
Exponent rules: (a^m)^n = a^(m*n)
Logarithmic inequality solving: a^x > b implies x > log_b(a)

Theorems

Exponentiation with same base
Logarithmic inequalities
Exponentiation rules for simplifying

Suitable Grade Level

Grades 9-12