Log 200 in terms of a (log 2) and b (log 3)?

[superjudge3]

If log 2 = a and log 3 = b, express log 200 in terms of a and b.

How does one go about solving this? Please show all steps. For those of you who don't realize, these are all log base 10. Thank you so much.

 

[pka]

\(\displaystyle \log(200)\) has no expression in terms of \(\displaystyle b=\log(3)\)!

Do you mean \(\displaystyle b=\log(5)\)?

 

[soroban]

Hello, superjudge3!

\(\displaystyle \text{If }\log 2 = a\,\text{ and }\log 3 = b\text{, express }\log 200\text{ in terms of }a\text{ and }b.\)

\(\displaystyle \text{We have: }\:\log(200) \;=\;\log(100\cdot2) \;=\;\log(100) + \log(2)\)

. . . . \(\displaystyle =\;\log(10^2) + \log(2) \;= \;2\underbrace{\log(10)}_{\text{This is 1}} + \underbrace{\log(2)}_{\text{This is }a} \;=\;2 + a\)

 

[superjudge3]

Than you soroban. Pka, the answer was 2+a. It said on the sheet (but of course, didn't explain why for us inquiring minds who got it wrong, so I decided to ask here and definetly will do so in the future).

 

[mmm4444bot]

… express log 200 in terms of a and b …

I would be interested in knowing why your instructor provided the answer 2 + a because this answer is not in terms of a and b.

:?

 

[fasteddie65]

Re:

… express log 200 in terms of a and b …

I would be interested in knowing why your instructor provided the answer 2 + a because this answer is not in terms of a and b.

:?

Actually, it is. Even though b is not used, no other variable except a is used in the answer. The directions did not say a number cannot be used. This is a perfectly consistent answer for this question.

 

[pka]

Actually, it is. Even though b is not used, no other variable except a is used in the answer. The directions did not say a number cannot be used. This is a perfectly consistent answer for this question.

fasteddie65, I actually agree with mmm4444bot on this point.
While you may be right for the general public, it does work in the strict community of mathematical question writers. This question would never have been accepted for a standard test. Read again the original question:

If log 2 = a and log 3 = b, express log 200 in terms of a and b.

If the suggested answer is to be correct then the question must be phrased as:
“If log 2 = a and log 3 = b, express log 200 in terms of a or b.”

 

[Deleted member 4993]

or the answer should be written as " 2 + a + 0*b"

 

[soroban]

\(\displaystyle \text{I suspect that it was a multi-part question,}\)

. . \(\displaystyle \text{that }b = \log3\text{ is used in parts (c), (d) and (e).}\)

 

[mmm4444bot]

I suspect that it [is] a multi-part question …

I do not think that other potential parts rectify anything with the instructions given for the posted part.

… the answer should be written as 2 + a + 0*b

Excellent! Subhotosh gets extra credit.