What is the value of arctanh(1/2) ?

The value of arctanh(1/2) is (ln(3))/2

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arctanh(1/2)

arctanh (\frac{1}{2})

\frac{ln ( 3 )}{2}

Decimal Notation

0.54930 \dots

Solution steps

arctanh (\frac{1}{2})

Use the Hyperbolic identity:

arctanh (x) = \frac{ln ( \frac{1 + x}{1 - x} )}{2}

= \frac{ln ( \frac{1 + \frac{1}{2}}{1 - \frac{1}{2}} )}{2}

\frac{ln ( \frac{1 + \frac{1}{2}}{1 - \frac{1}{2}} )}{2} = \frac{ln ( 3 )}{2}

\frac{ln ( \frac{1 + \frac{1}{2}}{1 - \frac{1}{2}} )}{2}

\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}} = 3

\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}}

Join

1 - \frac{1}{2} : \frac{1}{2}

1 - \frac{1}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 - 1}{2}

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1 + \frac{1}{2}}{\frac{1}{2}}

Join

1 + \frac{1}{2} : \frac{3}{2}

1 + \frac{1}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 + 1}{2}

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1

Add the numbers:

2 + 1 = 3

= 3

= \frac{3}{2}

= \frac{\frac{3}{2}}{\frac{1}{2}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{3 \cdot 2}{2 \cdot 1}

Refine

= \frac{3 \cdot 2}{2}

Cancel the common factor:

2

= 3

= \frac{ln ( 3 )}{2}

= \frac{ln ( 3 )}{2}

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