When evaluating exponential equations using logarithms, why do we need to isolate the variable first?

Question

For example,

 

3e^(2x)=1821

 

The correct solution would be,

 

1. Divide 3 on each side, thus         e^(2x)=607

2. Take the natural log on both sides:          lne^(2x)=ln607

3. Eliminate ln and e:           2x=ln607

4. Final solution is                x=ln607/2

 

However, how come we can't take the natural log on both sides first, before dividing 3?

 

1. Take the natural log of both sides:       ln3e^(2x)=ln1821

2. Use expanding property:                ln3 + lne^(2x) = ln1821

3. Eliminate lne and subtract ln3:              2x=ln1821-ln3

4. Use quotient rule for condensing and divide each side by 2:           x = ln(1821/3)/2

5. Final solution:              x = ln1818/2

 

 

Answer 1

Line 5 in your second solution should say

 

Final solution:  x = ln(1821/3) / 2

 

Both approaches are correct, and indeed give the same number, which you can easily verify.

 

ln(607) /  2 = ln(1821/3) / 2,

 

because 1821/3 = 607

 

But the first approach got you the answer in 3 steps.  The second approach got you the answer is 4 steps.  Which approach would YOU prefer to use in the real world, where you may have to do hundreds of calculations?