What is ln(e)+ln(sqrt(e))+ln(1)+ln(e^{ln(10)}) ?

The solution to ln(e)+ln(sqrt(e))+ln(1)+ln(e^{ln(10)}) is ln(10sqrt(e))+1

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simplify ln(e)+ln(sqrt(e))+ln(1)+ln(e^{ln(10)})

simplify

ln (e) + ln (e) + ln (1) + ln (e^{l n (10)})

ln (10 e) + 1

Decimal

3.80258 \dots

Solution steps

ln (e) + ln (e) + ln (1) + ln (e^{l n (10)})

Apply log rule:

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

ln (e) + ln (e^{l n (10)}) = ln (e e^{l n (10)})

= ln (e) + ln (e e^{l n (10)}) + ln (1)

e^{l n (10)} = 10

= ln (e) + ln (10 e) + ln (1)

Apply log rule:

lo g_{a} (a) = 1

= 1 + ln (10 e) + ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 1 + ln (10 e) + 0

Add the numbers:

1 + 0 = 1

= ln (10 e) + 1

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What is ln(e)+ln(sqrt(e))+ln(1)+ln(e^{ln(10)}) ?
The solution to ln(e)+ln(sqrt(e))+ln(1)+ln(e^{ln(10)}) is ln(10sqrt(e))+1