Question

3 power log root x at the base root 3 + 9 power log x at the base of 3 is

Answer 1

Answer:

3log3(x)+xlog3(x)=162

But  3log3(x)=x  and  log3(xlog3(x))=(logx(3))2 . Therefore  xlog3(x)=3(logx(3))2 .

Let  y=log3(x) , then  3y+3y2=162 . This clearly has two solutions,  y  a little more than  2  and  y  a little less than  −2 . I.e.  x  is a little more than  9  or  x  is a little less then  1/9 .

To solve this you will need to use numerical approximations. The simplest method is repeated bisection. Find  3y+3y2  when  y=2  and when  y=3 . One is smaller than  162 , and one is greater. Take the average. At each step you get an interval surrounding the answer. Keep bisecting the interval between the two closest approximations until the answer is accurate enough for your purposes.

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