[中四]對數函數

1.
log 2 + log 5 = log (2*5) = log 10 = 1

2.
3 log 2 + log 125 = log (2³) + log 125 = log (2³*125) = log (1000) = 3

3.
(log 32)/(log 4) = (log 2^5)/(log 2²) = (5 log 2)/(2 log 2) = 5/2

4.
log 1 * log 7 = 0 * log 7 = 0

5.
(log ³√3)/(log 3) = (log 3^(1/3))/(log 3) = 1/3* log 3 / log 3 = 1/3

6.
2 log x + log (1/x²) = log x² + log (1/x²) = log (x² * 1/x²) = log 1 = 0

7.
log √16 + log √325 - log √52
= log (√16 * √325 / √52)
= log √(16*325/52)
= log √100
= log 10
= 1

14a.
log_4 (12) + log_4 (9) - log_4 (108)
= log_4 (12*9/108)
= log_4 (1)
= 0

14b.
log 36 / log 216 = log 6² / log 6³ = (3 log 6)/(2 log 6) = 3/2

14c.
log_a [^5√(a^3)] + log_a [^4√(a^5)]
= log_a [a^(3/5)] + log_a [a^(5/4)]
= 3/5 + 5/4
= 37/20

14d.
[log_2 (6) + log_2 (3)]/[log_2 (36) + log_2 (9)]
= [log_2 (6*3)]/[log_2 (36*9)]
= log_2 (18) / log_2 (324)
= log_2 (18) / log_2 (18²)
= log_2 (18) / [2 log_2 (18)]
= 1/2

其實最重要的就是要知道對數的公式：
1. log a + log b = log (a*b)
2. log a - log b = log (a/b)
3. log m^n = n log m
4. log_a (a^n) = n

記熟了這些後，其他的都不太難了。
希望幫倒你！ ^^

(1)
log 2+log 5
=log (2．5)
=log 10
=1

(2)
3 log 2+log 125
=log (2^3)+log 125
=log 8+log 125
=log (8．125)
=log 1000
=3

(3)
log 32 / log 4
=log (2^5) / log (2^2)
=5 log 2 / 2 log 2
=5/2

(4)
log 1．log 7
=0．log 7
=0

(5)
log [(3)^(1/3)]/log 3
=log 3/3 log 3
=1/3

(6)
2 log x+log (1/x^2)
=2 log x+log [x^(-2)]
=2 log x-2 log x
=0

(7)
log [(16)^(1/2)]+log [325^(1/2)]-log [52^(1/2)]
=log (16．325 / 52) /2
=log 100 /2
=2/2
=1

(14)
底為a時以log_a 表示
(a)
log_4 (12)+log_4 (9) - log_4 (108)
=log_4 (12．9 / 108)
=log_4 (1)
=0

(b)
log 36/ log 216
=log (6^2) / log (6^3)
=2 log 6 / 3 log 6
=2/3

(c)
log_a [a^(3/5)+a^(5/4)]
=log_a [a^(3/5+5/4)]
=log_a [a^(37/20)]
=37(log _a a)/20
=37/20

(d)
[log_2 (6)+log_2 (3)]/[log_2 (36)+log_2 (9)]
=log_2 (6+3) / log_2 (36+9)
=log_2 (9) / log_2 (45)
=log_2 (9) / [log_2 (5)+log_2 (9)]

1. log2 + log5 = log(2x5) = log 10 = 1
2. 3log2 + log125 = log (3^2) + log 125 = log 8 + log 125 = log (8 x 125) = log 1000 = 3
3. log 32 / log 4 = log (2^5) / log (2^2) = (5log2)/(2log2) = 5/2
4. log 1 x log 7 = 0 x log 7 = 0
5. log [cubic root(3)] / log 3 = (1/3) log 3/ log3 = 1/3
6. 2 log x + log (1/x^2) = log x^2 + log (1/x^2) = log [(x^2)(1/x^2)] = log 1 = 0
7. log sqrt(16) + log sqrt(325) - log sqrt(52) = log [sqrt(16) x sqrt(325) / sqrt(52)]
= log [sqrt(16 x 325 / 52)] = log sqrt(100) = log 10 = 1

1 log 2+log 5
=log (2 x 5)
=log 10
=1

2 3 log 2 + log 125
=3 log 2 +log (5^3)
=3 log 2 + 3 log 5
=3(log 2 +log 5)
=3(log(2 x 5)
=3 x log 10
=3 x 1
=3

3 log 32 / log 4
=log (2^5)/log(2^2)
=5log 2 / 2log 2
=5/2
=2.5

4 log1 x log 7
=0 x log 7
=0

5 log cube root(3) / log3
= [1/3 (log3)]/log 3
=(1/3)/1
=1/3

6 2 log x + log (1/x²)
=2 log x + (1/2) log x
=log x (2+ 1/2)
=(5 log x)/2

7 log√16 + log√325- log√52
=log(√16 x √325 / √52)
=log √100
=log 10
=1

14a log(4)12+log(4)9-log(4)108
=log(4) x (12 x9 /108)
=log(4) 1
=0

14b log36 / log 216
=log(6^2) / log(6^3)
=2 log6 / 3 log 6
=2/3

14c 原式
=log(a) a^(3/5) +log(a) a^(5/4)
=log a^(3/5) / log a +log a^(5/4) / log a
= 3/5 log a / log a + 5/4 log a /log a
=3/5 +5/4
=1.85

14d log(2) 6 +log(2) 3 / log(2) 36 +log(2) 9
= [(log 6 +log 3)/log 2] / [(log 36 + log 9)/log 2)
=(log 18 / log 2 ) / [(log 324) / log 2]
=log 18 / log 324
=log 18 / 2 log 18
=1/2
=0.5