What is ln(e^{ln(x)}(e^{ln(ex)})) ?

The solution to ln(e^{ln(x)}(e^{ln(ex)})) is 2ln(x)+1

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simplify ln(e^{ln(x)}(e^{ln(ex)}))

simplify

ln (e^{l n (x)} (e^{l n (e x)}))

2 ln (x) + 1

Solution steps

ln (e^{l n (x)} (e^{l n (e x)}))

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (e^{l n (x)} (e^{l n (e x)})) = ln (e^{l n (x)}) + ln (e^{l n (e x)})

= ln (e^{l n (x)}) + ln (e^{l n (e x)})

Apply log rule:

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

ln (e^{l n (x)}) = ln (x) ln (e)

= ln (x) ln (e) + ln (e^{l n (e x)})

Apply log rule:

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

ln (e^{l n (e x)}) = ln (e x) ln (e)

= ln (x) ln (e) + ln (e x) ln (e)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (e x) ln (e) = ln (e) ln (e) + ln (e) ln (x)

= ln (e) ln (x) + ln (e) ln (e) + ln (e) ln (x)

Simplify

ln (x) ln (e) + ln (e) ln (e) + ln (e) ln (x) : 2 ln (e) ln (x) + ln (e) ln (e)

2 ln (e) ln (x) = 2 ln (x)

ln (e) ln (e) = 1

= 2 ln (x) + 1

lo g_{7} (x^{4} y^{6})

ln (- \frac{1}{π} ln (- 1))

lo g_{b} (2 x^{3} x^{4})

ln (\frac{30}{54})

ln (ln (2)) - ln (ln (4))

What is ln(e^{ln(x)}(e^{ln(ex)})) ?
The solution to ln(e^{ln(x)}(e^{ln(ex)})) is 2ln(x)+1