Question

Help please
2. Identify the equation(s) that are wrong. Explain any incorrect ones; provide complete details.

A. log 6(m + n) = log 6 • log (m + n)

B. log 10 = 1 ( I know this is correct)

C. log m/log n = log(m/n)

Answer 1

Apply this rule to the left side of A. What do you get?
\(\log ab = \log a + \log b\)

Answer 2

B. is correct as you stated.

Answer 3

C. \(\dfrac{\log a}{\log b} = \log a - \log b\)
Apply this rule to the left side of C.

Answer 4

A) log 6 m * log 6 n
C) log m - log n
Are these correct?

Answer 5

For A., the rule is the log of a product is the sum of the logs.
The product is 6 * (m + n), so
\(\log 6(m + n) = \log 6 + \log (m + n)\)
This is not the equation of problem A., so A. is incorrect.

Answer 6

If in A., you decide to multiply out (distribute) the 6, then you have:
\(\log (6m + 6n)\)
Now you have the log of a sum (the sum of 6m and 6n). There is no rule for the log of a sum.

Answer 7

In C. you are correct.
The equation should have been \(\log \dfrac{m}{n} = \log m - \log n\).

Answer 8

Thank you (A) was the most confusing for me but I think I understand what you explained. It would just be log 6 (m +n) = log 6 (m) + log 6(n)

Answer 9

No. That is still incorrect.

Answer 10

Remember that the rule of simplifying the log of a product only works if you take the log of a product.
Here are some simple examples:
\(\log ab = \log a + \log b\)
\(\log 6x = \log 6 + \log x\)
\(log (5 \times 10) = \log 5 + \log 10\)

Answer 11

Here is a slightly more complicated example:
\(\large \log [(x + 1)(x - 5)] = \log (x + 1) + \log (x - 5) \)

Answer 12

Notice that in all cases above, the log of the product was turned into the sum of the logs of the factors of the product.

Answer 13

Now let's go back to A. in your problem above.

Answer 14

This is what you were asked about. The question is whether it is a correct application of the rule of the log of a product or not.
A. log 6(m + n) = log 6 • log (m + n)

Answer 15

To find out, we apply the rule to the left side.
The left side is: \(\log 6(m+ n)\)
We need to take the log of the product of 6 and (m + n)

Answer 16

Since the rule is that the log of a product is the sum of the logs, then that means that
\(\log 6(m + n) = \log 6 + \log (m + n)\)
Notice on the right side, we have the sum of log 6 and log (m + n)
The problem is incorrect because is shows the product of log 6 and log (m + n)

Answer 17

Oh! ok I see what you mean thank you.

Answer 18

You're welcome.