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

a. combine the exponential expressions to a single expoential expression,

b. convert to logarthmic eqation.

c. solve equation

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To combine the expression, use the law of exponents,

e^(x)* e^(x+1)=1Gives, e^(x+x+1)=1 [Law: (x^m)* (x^n)= x^(m+n) ]=> e^(2x+1)=1Now to convert to log, take log of both the sides,gives, log(e^(2x+1)) = log(1)Using properties of logarithms,=> (2x+1)log(e)= 0 [Laws: log(x^a)= a log(x); log(1) =0 ]Now, to solve the equation, we have log(e)= 1,=> 2x+1 = 0=> x= -(1/2)

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

Gives, e^(x+x+1)=1 [Law: (x^m)* (x^n)= x^(m+n) ]

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

Now to convert to log, take log of both the sides,

gives, log(e^(2x+1)) = log(1)

Using properties of logarithms,

=> (2x+1)log(e)= 0 [Laws: log(x^a)= a log(x); log(1) =0 ]

Now, to solve the equation, we have log(e)= 1,

=> 2x+1 = 0

=> x= -(1/2)

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