What is the value of arctanh(1/3) ?

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

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arctanh(1/3)

arctanh (\frac{1}{3})

\frac{ln ( 2 )}{2}

Decimal

0.34657 \dots

Solution steps

arctanh (\frac{1}{3})

Use the Hyperbolic identity:

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

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

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

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

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

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

Join

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

1 - \frac{1}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 3 - 1 = 2

1 \cdot 3 - 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

= \frac{2}{3}

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

Join

1 + \frac{1}{3} : \frac{4}{3}

1 + \frac{1}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 3 + 1 = 4

1 \cdot 3 + 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 + 1

Add the numbers:

3 + 1 = 4

= 4

= \frac{4}{3}

= \frac{\frac{4}{3}}{\frac{2}{3}}

Divide fractions:

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

= \frac{4 \cdot 3}{3 \cdot 2}

Cancel the common factor:

3

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

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

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