What is expand ln((1/5)^{ln(x)}) ?

The solution to expand ln((1/5)^{ln(x)}) is -ln(5)ln(x)

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expand ln((1/5)^{ln(x)})

expand

ln ((\frac{1}{5})^{l n (x)})

- ln (5) ln (x)

Solution steps

ln ((\frac{1}{5})^{l n (x)})

Apply log rule:

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

ln ((\frac{1}{5})^{l n (x)}) = ln (x) ln (\frac{1}{5})

= ln (x) ln (\frac{1}{5})

Simplify

ln (\frac{1}{5}) : - ln (5)

= (- ln (5)) ln (x)

Remove parentheses:

(- a) = - a

= - ln (x) ln (5)

5 lo g_{9} (x) + 7 lo g_{9} (y) - 3 lo g_{9} (z)

- 3 ln (x + 2) + 4 ln (x + 3)

- 4 (ln (4 - v) - ln (4 - u)) + 4 (ln (v) - ln (u))

ln (x + 1) + ln (x^{2} + 1)

ln (x + 6) - ln (x)

What is expand ln((1/5)^{ln(x)}) ?
The solution to expand ln((1/5)^{ln(x)}) is -ln(5)ln(x)