What is the value of arctanh(0.8) ?

The value of arctanh(0.8) is ln(3)

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arctanh(0.8)

arctanh (0.8)

ln (3)

Decimal

1.09861 \dots

Solution steps

arctanh (0.8)

= arctanh (\frac{4}{5})

Rewrite using trig identities:

ln (3)

arctanh (\frac{4}{5})

Use the Hyperbolic identity:

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

= \frac{ln ( \frac{1 + \frac{4}{5}}{1 - \frac{4}{5}} )}{2}

\frac{ln ( \frac{1 + \frac{4}{5}}{1 - \frac{4}{5}} )}{2} = ln (3)

\frac{ln ( \frac{1 + \frac{4}{5}}{1 - \frac{4}{5}} )}{2}

\frac{1 + \frac{4}{5}}{1 - \frac{4}{5}} = 9

\frac{1 + \frac{4}{5}}{1 - \frac{4}{5}}

Join

1 - \frac{4}{5} : \frac{1}{5}

1 - \frac{4}{5}

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= \frac{1 \cdot 5}{5} - \frac{4}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 5 - 4}{5}

1 \cdot 5 - 4 = 1

1 \cdot 5 - 4

Multiply the numbers:

1 \cdot 5 = 5

= 5 - 4

Subtract the numbers:

5 - 4 = 1

= 1

= \frac{1}{5}

= \frac{1 + \frac{4}{5}}{\frac{1}{5}}

Join

1 + \frac{4}{5} : \frac{9}{5}

1 + \frac{4}{5}

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= \frac{1 \cdot 5}{5} + \frac{4}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 5 + 4}{5}

1 \cdot 5 + 4 = 9

1 \cdot 5 + 4

Multiply the numbers:

1 \cdot 5 = 5

= 5 + 4

Add the numbers:

5 + 4 = 9

= 9

= \frac{9}{5}

= \frac{\frac{9}{5}}{\frac{1}{5}}

Divide fractions:

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

= \frac{9 \cdot 5}{5 \cdot 1}

Refine

= \frac{9 \cdot 5}{5}

Cancel the common factor:

5

= 9

= \frac{ln ( 9 )}{2}

Simplify

ln (9) : 2 ln (3)

ln (9)

Rewrite

9

in power-base form:

9 = 3^{2}

= ln (3^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (3^{2}) = 2 ln (3)

= 2 ln (3)

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

Divide the numbers:

\frac{2}{2} = 1

= ln (3)

= ln (3)

= ln (3)