Verified answer

1. Well, first you have to understand that ln e= 1, because the base of the natural logrithim is e.

So ln(e^e^10)=

ln(e^10 ln e), remember, the exponent goes in front of the log/ln symbol.

ln(e^10 * 1), ln e=1, remember that.

ln e^10, just get rid of the 1 because of the identity property of multiplication.

10 ln e, exponents goes before the log/ln symbol.

10, ln e=1.

So the exact value of that long expression is 10.

2. ln (1+x^2) + 1/2 ln x- ln sinx

ln (1+x^2) + ln x^(1/2) - ln sinx

ln (1+x^2) + ln sqrt. x - ln sinx

ln sqrt. x(1+x^2) - ln sin x

ln (sqrt. x(1+x^2)) / (sinx(

3. (sin^-1) (sin (7pi/3))

Convert 7pi/3 into degrees or locate them in the unit circle. It is 420 degrees and located in quadrant I. Draw a reference angle by subtracting 420-360= 60 degrees. The coordinate of 60 degrees in the unit circle is (1/2, sqrt3/2). The sin value is sqrt. 3/2. If you find the inverse sine of this, it should be 60 degrees.

:D

e^10

ln [ sqrt ( x ) ( 1 + x^2 ) / sin x ]

7pi / 3