Calculating the Ratio of Line Segments Using Distance Formula

Use the distance formula between two points:

Distance = √((x₂ – x₁)² + (y₂ – y₁)²)

For points (1, 2) and (3, 4):

Length₁ = √((3 – 1)² + (4 – 2)²)

Simplify the differences:

Length₁ = √(2² + 2²)

Calculate the squares:

Length₁ = √(4 + 4)

Add the results:

Length₁ = √8

Approximate the square root:

Length₁ ≈ 2.828

Use the distance formula again for points (1, 2) and (-1, -3):

Length₂ = √((-1 – 1)² + (-3 – 2)²)

Simplify the differences:

Length₂ = √((-2)² + (-5)²)

Calculate the squares:

Length₂ = √(4 + 25)

Add the results:

Length₂ = √29

Approximate the square root:

Length₂ ≈ 5.385

Use the formula for the ratio of the two lengths:

Ratio = Length₂ ÷ Length₁

Substitute the approximate values:

Ratio = 5.385 ÷ 2.828

Perform the division:

Ratio ≈ 1.904

Thus, the approximate ratio of the lengths is 1.904.

What fraction of x should a be to make x equals 3 the solution to x=ln(1/(ax))

To find the fraction of x that a must be to ensure the solution is 3, consider this:

  1. Start with x = ln(1/(ax)). Set x = 3 to solve for a.
  • Rewrite ln(1/(ax)) in exponential form:
    e^x = 1/(ax).
  • Isolate a:
    a = 1/(x * e^x).
  • With x = 3, the value for a is:
    a = 1/(3 * e^3).
  • To get the fraction that a represents of x:
    fraction = 1/(9 * e^3).

Thus, the fraction of x that a must be to ensure the solution is 3 is 1/(9 * e^3).

Log Equations Quizzes

Welcome to your log equations

solve this log equation

domain of 2 lnx versus the domain of ln of x squared are they the same

Domain of ln(x²):
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The argument of the logarithm function x² must be positive. This is true for any x except zero, so the domain of ln(x²) is all real numbers except zero: x ∈ (−∞, 0) ∪ (0, +∞).

Example:
– For x = −2, x² = 4. ln(4) is valid.
– For x = 3, x² = 9. ln(9) is valid.
– For x = 0, x² = 0 and ln(0) is not defined.

Domain of 2ln(x):
——————————
The argument x must be positive, so the domain is only positive real numbers: x ∈ (0, +∞).

Example:
– For x = 1, 2ln(1) = 2 × 0 = 0.
– For x = 5, 2ln(5) is valid.
– For x = −3 or x = 0, 2ln(−3) and 2ln(0) are not defined.

Conclusion:
——————————
ln(x²) is defined for any nonzero x, while 2ln(x) is only defined for positive x. Despite their algebraic similarities, they are distinct in terms of domain. ln(x²) can handle negative inputs because squaring them results in a positive number suitable for the log function. In contrast, 2ln(x) does not permit negative or zero inputs.

3-Vector Dot Product Calculator: Master the Math with Decimals

Acceptable Input Formats

This calculator accepts decimal numbers in the following formats:

  • Whole Numbers: e.g., 1, 2, 3
  • Decimal Numbers: e.g., 1.5, 2.75, 3.333
  • Negative Numbers: e.g., -1, -1.5, -2.75
  • Zero: 0

Please note that fractions (e.g., 1/2, 2/3) are not supported and should be converted to decimal form.

3-Vector Dot Product Calculator

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