ln e^x + ln e^3 = ln e^5

To find x values, we will use the natural logarithm properties:

We know that: ln a^b = b*ln a

==> x*lne + 3*ln e = 5*ln e

Also from the logarithm properties we know that :

ln e = lne (e) = 1

==> x*1 + 3*1 = 5*1

==> x + 3 = 5

Now subtract 3 from both sides:

==> x + 3 - 3 = 5 - 3

**==> x = 2**

Then the answer is :

**x = 2**

To check, we will substitute:

ln e^2 + ln e^3 = ln e^5

2lne + 3ln e = 5 lne

==> 2 + 3 = 5

==> 5= 5

We have to find x given ln e^x + ln e^3 = ln e^5.

We use the relation log a^b = b*log a and log ab = log + log b.

ln e^x + ln e^3 = ln e^5

=> ln e^x * e^3 = ln e^5

=> ln e^ ( x +3) = ln e^5

take the antilog of both the sides

=> x + 3 = 5

=> x = 2

ln e^x is defined for x = 2.

**Therefore the required value of x is 2. **

Given lne^x+lne^3= lne^5.

We have to determine x.

We know by definition of ln , the natural logarithms, that

If e^a = b, then a = lnb.

Therfore ln e^a = c implies e^a = e^c implies a =c or a = ln e^a.

So by definition , ln e^a = a.

Therefore the given equation becomes:

x+3 = 5.

Subtract 3:

x+3-3 = 5-3.

x = 2.

Therfore the solution of ln e^x+lne^3 = ln 5 is x = 2.