1966 AMC 12 第 12 题

Question

满足方程的实数

x

的个数

(2^{6x+3})(4^{3x+6})=8^{4x+5}

为：

The number of real values of

x

that satisfy the equation

(2^{6x+3})(4^{3x+6})=8^{4x+5}

is:

0

1

2

3

大于

3

greater than

3

将每一项都改写成以

2

为底

Rewrite every term with base

2

在求

x

之前，先比较等式两边的总指数

Compare the total exponent on each side before trying to solve for

x

改写成以

2

为底后，左边的指数为

\begin{aligned} &(6x+3)+2(3x+6)\\ &\qquad=12x+15 \end{aligned}\text{，}

而右边的指数为

3(4x+5)=12x+15

。该方程对每个实数

x

都是恒等式，所以实数解多于三个。

因此，正确答案是 E。

In base

2,

the left exponent is

\begin{aligned} &(6x+3)+2(3x+6)\\ &\qquad=12x+15, \end{aligned}

and the right exponent is

3(4x+5)=12x+15.

The equation is an identity for every real

x,

so it has more than three real solutions.

Therefore, the correct answer is E.

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