Question

有些对数问题 请帮忙看下

a=lg5 lg20=?(用a表示) 如果lgab=4 那么lgalgb的最大值为? 已知函数f(x)=x平方lga+2x+4lga的最大值为3 则实数a=?

Answer 1

1,
a
=
lg5,
a
+
lg2
=
lg5
+
lg2
=
lg10
=
1,
lg2
=
1
-
a,
lg20
=
lg5
+
2lg2
=
a
+
2(1-a)
=
2
-
a.
2,
lg(ab)
=
lga
+
lgb
=
4,
lgalgb
<=
[lga
+
lgb]^2/4
=
4^2/4
=
4,
当lga
=
lgb
=
2,
a
=
b
=
10^2
=
100时，lgalgb达到最大值
=
4.
3,
f(x)
=
x^2lga
+
2x
+
4lga的最大值为3，
因此，lga
<
0.
0
<
a
<
1.
f(x)的对称轴位于
x
=
-1/lga,
3
=
f(-1/lga)
=
(-1/lga)^2(lga)
+
2(-1/lga)
+
4lga
=
1/lga
-2/lga
+
4lga
=
4lga
-1/lga,
4(lga)^2
-
3lga
-
1
=
[4lga
+
1][lga
-
1]
=
0,
lga
=
1(不符题意，舍去)或者lga
=
-1/4,
a
=
10^(-1/4).