Solve the exponential equation 32x+3x−2=0. Express the solution set in terms of natural logarithms or common logarithms.Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.

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Laith Petty · Accepted Answer

Step 1  Given:  32x+3x−2=0  Step 2  Consider,  32x+3x−2=0 we write a given expression as:  (3x)2+3x−2=0  put 3x=y then (3x)2+3x−2=0 converted into quadratic equation given as:  y2+y−2=0  y2−y+2y−2=0  y(y-1)+2(y-1)=0  (y+2)(y-1)=0  y+2=0    or  y−1=0  y=−2    or  y=1  Step 3  Consider y=−2 but we have y=3x resubstitute as y=3x then we get  3x=−2  Taking both sides natural logarithm we get  ln⁡(3x)=log⁡(−2)  which is not possible because ln⁡(−2) not well defined.  If we consider common logarithm we get   ln⁡(3x)=log⁡(−2) which is not possible because log⁡(−2) not well defined.  Hence for3x=−2 is not possible solution for given expression.  Step 4  Consider y=1 but we have y=3x  resubstitute as y=3x then we get  3x=1  Taking both sides natural logarithm we get  ln⁡(3x)=ln⁡(1)  xln⁡(3)=0  x=0  If we consider common logarithm we get   log⁡(3x)=log⁡(1)  xlog⁡(3)=0   x=0  Hence  the solution set in terms of natural logarithms and  common logarithms for  32x+3x−2=0  is  {xx=0}

Solve the exponential equation 3^{2x}+3^x-2=0. Express the solution set in

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