求满足等式m - nlog₃2 = 10log₉6的整数m和n的值

Hey there! Let's work through this logarithm equation step by step to find integer values of m and n. The key here is to unify the base of all logarithms so we can compare like terms easily—since we have $\log_3$ on the left and $\log_9$ on the right, let's convert everything to base 3 first.

Step 1: Convert the right-hand side to base 3 #

Remember the logarithm power rule: $\log_{b^k}a = \frac{1}{k}\log_b a$. Since 9 is $3^2$, we can rewrite $\log_9 6$ as:
$$\log_9 6 = \log_{3^2}6 = \frac{1}{2}\log_3 6$$
Multiply this by 10 (as in the original equation):
$$10\log_9 6 = 10 \times \frac{1}{2}\log_3 6 = 5\log_3 6$$

Step 2: Expand $\log_3 6$ using the logarithm product rule #

The product rule states $\log_b(xy) = \log_b x + \log_b y$. Since 6 = 2×3, we can split the log:
$$\log_3 6 = \log_3(2 \times 3) = \log_3 2 + \log_3 3$$
We know $\log_3 3 = 1$, so this simplifies to:
$$\log_3 6 = 1 + \log_3 2$$
Substitute back into the right-hand side:
$$5\log_3 6 = 5(1 + \log_3 2) = 5 + 5\log_3 2$$

Step 3: Rewrite the original equation and group like terms #

Now our equation looks like this:
$$m - n\log_3 2 = 5 + 5\log_3 2$$
Let's rearrange terms to collect constants on one side and log terms on the other:
$$m - 5 = n\log_3 2 + 5\log_3 2$$
Factor out $\log_3 2$ on the right:
$$m - 5 = (n + 5)\log_3 2$$

Step 4: Use the irrationality of $\log_3 2$ to find integers m and n #

Here's the crucial point: $\log_3 2$ is an irrational number (if it were rational, say $\frac{p}{q}$, then $3^p = 2^q$, which is impossible since 3 and 2 are coprime—only possible if p=q=0, which doesn't fit $\log_3 2$).

Now, the left side $m-5$ is an integer, and the right side is an integer $(n+5)$ multiplied by an irrational number. The only way this equality holds is if the coefficient of the irrational term is zero (otherwise, the right side would be irrational, which can't equal an integer).

So:

1. Set the coefficient to zero: $n + 5 = 0$ → $n = -5$
2. Then the left side must also be zero: $m - 5 = 0$ → $m = 5$

Step 5: Verify the solution #

Let's plug m=5 and n=-5 back into the original equation to check:

• Left-hand side: $5 - (-5)\log_3 2 = 5 + 5\log_3 2$
• Right-hand side: $10\log_9 6 = 5\log_3 6 = 5(1 + \log_3 2) = 5 + 5\log_3 2$

Both sides match, so our solution is correct!

内容的提问来源于stack exchange，提问作者Fluellen