a) e^x -2e^(-x) = 1

Rewriting that in fraction form,

e^x -2/(e^x) = 1

Clear the fraction, multiply both sides by e^x,

(e^x)^2 -2 = e^x

(e^x)^2 -e^x -2 = 0

(e^x -2)(e^x +1) = 0

e^x = 2 or -1

When e^x = 2,

x*ln(e) = ln(2)

x = ln(2) ---------------------**

When e^x = -1

x*ln(e) = ln(-1)

Uh-oh, there is no log of negative numbers, so disregard e^x = -1

Therefore, x = ln(2) ------------answer.

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b) ln(x^2 +e) -ln(x +1) = 1

ln[(x^2 +e) /(x +1)] = ln(e)

(x^2 +e) /(x +1) = e

x^2 +e = e(x+1)

x^2 +e = ex +e

x^2 = ex

x^2 -ex = 0

x(x -e) = 0

x = 0 or e. ----------------answer.