The number of positive integers satisfying `x+(log)_(10)(2^x+1)=x(log)

1.	The number of positive integers satisfying `x+(log)_(10)(2^x+1)=x(log)_(10)5+(log)_(10)6`is...........
Answer» `x+log_10(2^x+1)=xlog_10 5+log_10 6` `log_10 10^x=xlog_10 10=lambda` `log_10 10^lambda+log_10(2^lambda+1)=log_10 5^x+log_10 6` `2^x5^x[2^x+1]=65^x=5^lambda[2^(2lambda(+2^lambda-6))]=0` `5^x=0,2^(2lambda)+2^lambda-6=0` `2^x=t^2+t-6=0` `t^2+3t-2t-6=0` `t(t+30-2(t+3)=0` `(t-2)(t+3)=0` `t=2,-3` `2^lambda-2=0` `lambda=1`.

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