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  1. 1Sine of 7π/6
  2. 2Secant of 2π/3
  3. 3Cosine of 3π/4
  4. 4Degrees to radians
  5. 5Radians to degrees
  6. 6Arc length from radians
  7. 7cosθ and tanθ from sinθ
  8. 8Double angle from sinθ
  9. 9cos75° via sum formula
  10. 10Solving 2sinx−1=0
  11. 11Solving 2cos²x−cosx−1=0
  12. 12Solving tanx=√3
  13. 13Solving sin2x=cosx
  14. 14Law of sines for a side
  15. 15Law of cosines, finding a side
  16. 16Largest angle, law of cosines
  17. 17Area with two sides and angle
  18. 18Inverse function value
  19. 19Inverse sine and cosine angles
  20. 20Evaluating log expressions
  21. 21Logs: solving log equation
  22. 22Solving ln(2x+1)=3
  23. 23Solving 2^(x+1)=3^x
  24. 24Solving exponential growth
  25. 25Solving a cubic by factoring
  26. 26Remainder theorem
  27. 27Roots of a cubic
  28. 28Asymptotes of a rational function
  29. 29Hole vs vertical asymptote
  30. 30Arithmetic sequence term and sum
  31. 31Geometric sequence, 8th term and sum
  32. 32Sum of infinite geometric series
  33. 33Vector magnitude and dot product
  34. 34Angle between two vectors
  35. 35Resolving a force vector
  36. 36Multiplying complex numbers
  37. 37Dividing complex numbers
  38. 38Quadratic with complex roots
  39. 39Modulus and argument of z

23Solving 2^(x+1)=3^x

Solve 2x+1=3x\displaystyle 2^{x + 1} = 3^x. Give the answer to 4 decimal places.

The solution is x ≈ 1.7095. Taking ln of both sides turns the exponents into multipliers: (x+1)ln2 = x ln3, then collecting x terms gives x = ln2/(ln3−ln2) ≈ 0.6931/0.4055 ≈ 1.7095. Check: both 2^2.7095 and 3^1.7095 ≈ 6.541.

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