ln e^3 + ln 6x = 6

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ln e^3 + ln 6x = 6

We know that ln e^a = alne = a

==> ln e^3 + ln 6x =6

==> 3ln e + ln 6x = 6

==> 3 + ln 6x =6

==> ln 6x = 3

==> 6x = e^3

==> x = e^3 /6

To solve lne^3 +ln6x =6

Solution:

6 = ln e^6. Therefore,

ln e^3 +ln6x = lne^6

ln {(e^3)6x)} = ln e^6. Take anti logarithms on both sides.

e^3* 6x = e^6

6x = e^6 / e^ 3 = e^3

6x = e^3

6x/6 = e^3/6

x = (e^3)/6

We'll use the product property of the logarithms:

ln e^3 + ln 6x = ln (6x*e^3)

We'll substitute the sum by the product:

ln (6x*e^3) = 6

So, we can write the expression above:

6x*e^3 = e^6

We'll divide by e^3 both sides:

6x = e^3

We'll divide by 6 both sides:

x = e^3/6

Since the result is positive, the equation has the solution

**x = e^3/6**

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