What is ln(3e)+ln(e^5)+ln(e) ?

The solution to ln(3e)+ln(e^5)+ln(e) is 7+ln(3)

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simplify ln(3e)+ln(e^5)+ln(e)

simplify

ln (3 e) + ln (e^{5}) + ln (e)

7 + ln (3)

Decimal Notation

8.09861 \dots

Solution steps

ln (3 e) + ln (e^{5}) + ln (e)

Apply log rule:

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

ln (3 e) = ln (3) + ln (e)

= ln (3) + ln (e) + ln (e^{5}) + ln (e)

Simplify

ln (3) + ln (e) + ln (e^{5}) + ln (e) : 2 ln (e) + ln (3) + ln (e^{5})

Apply log rule:

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

ln (e^{5}) = 5 ln (e)

= 2 ln (e) + ln (3) + 5 ln (e)

Group like terms

= 2 ln (e) + 5 ln (e) + ln (3)

Add similar elements:

2 ln (e) + 5 ln (e) = 7 ln (e)

= 7 ln (e) + ln (3)

7 ln (e) = 7

= 7 + ln (3)

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