Logarithmic function

(a) 1 + ln (e2x - 1) = ln (ex + 1)

1 + ln [(ex - 1)(ex + 1)] = ln (ex + 1)

1 + ln (ex - 1) + ln (ex + 1) = ln (ex + 1)

1 + ln (ex - 1) = 0

ln (ex - 1) = - 1

ex - 1 = e-1

ex = 1 + e-1

x = ln (1 + e-1)

(b) 2ex - 2e-x = 3

2(ex)2 - 2 = 3ex

2(ex)2 - 3ex - 2 = 0

(2ex + 1)(ex - 2) = 0

ex = -1/2 (rej) or 2

x = ln 2

(3) 32y - 32y-2 = 648

9 x 32y - 32y = 5832

32y = 729

2y = 6

y = 3

a) 1+log(e^2x -1)= log(e^x +1)
log10 + log (e^2x -1)= log(e^x +1)
log10(e^2x -1)= log(e^x +1)
10(e^2x -1)= e^x +1
10e^2x - e^x -11= 0
(10e^x - 11)(e^x + 1) = 0
e^x = 11/10
x = 0.0953
for e^x = -1
no solution