设函数f（x）=|lg（x+1）|，实数a，b（a＜b）满足f(a)=f(-b+1b+2)，f（10a+6b+21）=4lg2，求a，b的值．

Question

设函数f（x）=|lg（x+1）|，实数a，b（a＜b）满足f(a)=f(-(b+1)/(b+2))，f（10a+6b+21）=4lg2，求a，b的值．

Answer 1

因为f(a)=f(-(b+1)/(b+2))，所以|lg(a+1)|=|lg(-(b+1)/(b+2)+1)|=|lg((1)/(b+2))|=|lg(b+2)|，
所以a+1=b+2，或（a+1）（b+2）=1，又因为a＜b，所以a+1≠b+2，所以（a+1）（b+2）=1．
又由f（a）=|lg（a+1）|有意义知a+1＞0，从而0＜a+1＜b+1＜b+2，
于是0＜a+1＜1＜b+2．
所以(10a+6b+21)+1=10(a+1)+6(b+2)=6(b+2)+(10)/(b+2)＞1．
从而f(10a+6b+21)=|lg[6(b+2)+(10)/(b+2)]|=lg[6(b+2)+(10)/(b+2)]．
又f（10a+6b+21）=4lg2，
所以lg[6(b+2)+(10)/(b+2)]=4lg2，
故6(b+2)+(10)/(b+2)=16．解得b=-(1)/(3)或b=-1（舍去）．
把b=-(1)/(3)代入（a+1）（b+2）=1解得a=-(2)/(5)．
所以 a=-(2)/(5)，b=-(1)/(3)．