对数能力测试卷 7

正确选项：C

解决方案

已知,

a^2 + b^2 = c^2

⇒ b^2 = c^2 - a^2

⇒ b^2 = (c + a)(c - a) … (1)

现在，

1/log(c + a)b + 1/log(c-a)b

= logb(c + a) + logb(c - a) …. [logab = 1/logba]

= logb((c + a) * (c - a)) [log(m) + log(n) = log(m * n)]

= logb(c^2 - a^2)

= logb(c + a)(c - a)

= logb b^2 …. (从 (1) 式)

= 2 logb b

= 2

正确选项：C

解决方案

已知,

log(x + 5) + log(x - 5) = 4log2 + 2log3

⇒ log(x + 5)(x - 5) = 4log2 + 2log3 …. [log(m) + log(n) = log(m * n)]

⇒ log(x^2 - 25) = log2^4 + log3^2 …. [along(b) = log(b^a)]

⇒ log(x^2 - 25) = log16 + log9

⇒ log(x^2 - 25) = log(16×9)

⇒ log(x^2 - 25) = log144

⇒ x^2 - 25 = 144

⇒ x^2 = 169

⇒ |x| = √169

⇒ x = 13 或 x = -13

当 x = -13 时，(x + 5) 和 (x - 5) 将变为负数，而对数对于负数没有定义，因此我们将舍弃 x = -13

因此，x = 13

正确选项：A

解决方案

log x = (log 196/log14)

⇒ log x = [log(14×14)/log14]

⇒ log x = log 14^2/log 14

⇒ log x = 2log 14/log 14 …. [log(a^2) = 2log(a)]

⇒ log x = 2

此处底数为 10

⇒ log10x = 2

⇒ x = 10^2 …. [logac = b ⇒ a^b = c]

⇒ x = 10×10

x = 100

正确选项：C

解决方案

log100.000001

= log1010^(-6)

= log101/10^(6)

= log10 (1) - log10 (10^6) …. [log(a/b) = log(a) - log(b)]

= 0 - log10 (10^6) …. [log(1) 对于所有底数都等于 0。]

= 0 - 6 log10 (10) …. [log(a^b) = b log(a)]

= 0 - 6 * 1 …. [log10 (10) = 1]

= - 6

正确选项：A

解决方案

已知,

[log10 (5 log10100)]^2

= [log10 (5 log1010^2)]^2

= [log10 (5 * 2log1010)]^2 …. [log(a^b) = b log(a)]

= [log10 (10 * log1010)]^2

= [log10 (10 * 1)]^2 …. [log10 (10) = 1]

= 1^2

= 1