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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2000

  • question_answer
    If \[\log 2,\log ({{2}^{x}}-1)\]and \[\log ({{2}^{x}}+3)\]are in A.P., then \[x\] is equal to:

    A)  5/2                                       

    B)  log 25

    C)  log 23                                  

    D)  logs 2

    Correct Answer: B

    Solution :

    \[\log 2,\log ({{2}^{x}}-1),\log ({{2}^{x}}+3)\]are in A.P\[2,({{2}^{x}}-1),({{2}^{x}}+3)\]are in G.P Put \[{{2}^{x}}=t\] \[2,(t-1)(t+3)\]are in G.P \[{{(t-1)}^{2}}=2(t+3)\]\[\Rightarrow \]\[{{t}^{2}}+1-2t=2t+6\] \[{{t}^{2}}-4t-5=0\]   \[\Rightarrow \]\[{{t}^{2}}-5t+t-5=0\] \[t(t-5)+1(t-5)=0\]\[\Rightarrow \]\[t=-1,5\] \[\Rightarrow \]\[t\ne -1\]                 \[{{2}^{x}}=5\]or \[x={{\log }_{2}}5\]


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