高中數學對數比大小

Question

1.設a,b,c,d為正實數,若2^a=3^b=log2 c=log3 d>0,a,b,c,d比大小

2.設a>b>c>d>1且b^2=ac,試比較loga b,logb c,logc a之大小

已更新項目:

請問可梅k=2^a > 1

應該k>0

2 個已更新項目:

k>0設k=0.1則2^a<1,如何判斷c>a

Answer 1

最佳解答

1.

0< log2 < log3 < 1,

設2^a = 3^b = log_2 (c)= log_3(d) = k, 則

k=2^a > 1

a=logk/log2

b=logk/log3

c= 2^k

d= 3^k

故 d > c > a > b

2.

b²=ac, 故 c, b, a為等比,且公比 > 1, 設 c=k >1, b=kr, a=kr², logk > 0, logr>0

log_a(b)=log(b) / log(a)= (logk+ logr)/(logk+2logr)= 1- logr/(logk+2logr)

log_b(c)=log(c) / log(b)= (logk)/(logk+logr)= 1 - logr/(logk+logr)

log_c(a)=log(a) / log(c)= (logk+ 2logr)/(logk)= 1+ 2logr / logk

故log_c (a) > log_a (b) > log_b (c)

2011-12-08 20:42:18 補充：

a > 0, k=2^a > 1