Question

有多少个正整数 $b$ 满足 $\log_b 729$ 是正整数？

How many positive integers $b$ have the property that $\log_b 729$ is a positive integer?

$0$

$1$

$2$

$3$

$4$

若 $\log_b 729 = n$ ，则 $b^n = 729 = 3^6$ ，所以 $n$ 必须是 $6$ 的正因数。

$n = 1, 2, 3, 6$ 分别给出 $b = 3^6, 3^3, 3^2, 3^1$ 也就是 $729, 27, 9$ ，和 $3$ 。

共有 $4$ 个这样的 $b$ 。

因此，正确答案是 E。

If $\log_b 729 = n,$ then $b^n = 729 = 3^6,$ so $n$ must be a positive divisor of $6.$

The possibilities $n = 1, 2, 3, 6$ give $b = 3^6, 3^3, 3^2, 3^1,$ that is, $729, 27, 9,$ and $3.$

There are $4$ such values of $b.$

Thus, the correct answer is E.