What is expand ln((1/x)^{1/(ln(x))}) ?

The solution to expand ln((1/x)^{1/(ln(x))}) is -1

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expand ln((1/x)^{1/(ln(x))})

expand

ln ((\frac{1}{x})^{\frac{1}{l n ( x )}})

- 1

Solution steps

ln ((\frac{1}{x})^{\frac{1}{l n ( x )}})

Apply log rule:

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

ln ((\frac{1}{x})^{\frac{1}{l n ( x )}}) = \frac{1}{ln ( x )} ln (\frac{1}{x})

= \frac{1}{ln ( x )} ln (\frac{1}{x})

Multiply fractions:

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

= \frac{1 \cdot ln ( \frac{1}{x} )}{ln ( x )}

Multiply:

1 \cdot ln (\frac{1}{x}) = ln (\frac{1}{x})

= \frac{ln ( \frac{1}{x} )}{ln ( x )}

Simplify

ln (\frac{1}{x}) : - ln (x)

= \frac{- ln ( x )}{ln ( x )}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{ln ( x )}{ln ( x )}

Apply rule

\frac{a}{a} = 1

= - 1

ln (\frac{486}{271})

\frac{4 ln ( 2 )}{ln ( \frac{512}{3} )}

ln (\frac{400}{100})

ln (\frac{4421}{28350})

- lo g_{10} (4.6 \cdot 1 0^{- 7})

What is expand ln((1/x)^{1/(ln(x))}) ?
The solution to expand ln((1/x)^{1/(ln(x))}) is -1