1951 AMC 12 第 45 题

Question

已知

\log 8=0.9031

和

\log 9=0.9542

，则下列唯一不能在不查表的情况下求出的对数是：

If you are given

\log 8=0.9031

and

\log 9=0.9542,

then the only logarithm that cannot be found without the use of tables is:

\log 17

\log(\frac{5}{4})

\log 15

\log 600

\log 0.4

两个已知值可以确定

\log2

与

\log3

The two given values determine

\log2

and

\log3

使用

\log5=1-\log2

，再判断哪个选项含有

2,3,5

以外的质因数

Use

\log5=1-\log2

, then see which choice has a prime factor other than

2,3,5

由已知条件，

\begin{gathered} \log2=\frac13\log8,\\ \log3=\frac12\log9 \end{gathered}\text{，}

并且

\log5=1-\log2

。因此，可以求出由

2,3,5

的幂相乘或相除所得各数的对数。

17

不含这些质因数中的任何一个，所以仅凭已知条件不能确定其对数。

因此，正确答案是 A。

From the data,

\begin{gathered} \log2=\frac13\log8,\\ \log3=\frac12\log9, \end{gathered}

and

\log5=1-\log2.

Therefore logarithms of products and quotients of powers of

2,3,5

can be computed. The number

17

has none of those prime factors, so its logarithm is not determined by the data.

Thus, the correct answer is A.

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