Difference between revisions of "2020 AMC 12B Problems/Problem 13"
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~ TheBeast5520 | ~ TheBeast5520 | ||
− | == Solution 5 ( | + | == Solution 5 (Solution 1 but more detailed)== |
Note: Only use this method if all else fails and you cannot find a way to simplify the logarithms. | Note: Only use this method if all else fails and you cannot find a way to simplify the logarithms. | ||
+ | |||
+ | We can see that <math>\log_2{6}</math> is greater than <math>2</math> and less than <math>3</math>. Additionally, since <math>6</math> is halfway between <math>2^2</math> and <math>2^3</math>, knowing how exponents increase more the larger <math>x</math> is, we can deduce that <math>\log_2{6}</math> is just above halfway between <math>2</math> and <math>3</math>. We can guesstimate this as <math>\log_2{6} \approx 2.55.</math> (It's actually about <math>2.585.</math>) | ||
+ | |||
+ | Next, we think of <math>\log_3{6}.</math> This is greater than <math>1</math> and less than <math>2</math>. <math>6</math> is halfway between <math>3^1</math> and <math>3^2,</math> and similar to the logic for <math>\log_2{6},</math> we know that <math>\log_3{6}</math> is just above halfway between <math>1</math> and <math>2</math>. We guesstimate this as <math>\log_3{6} \approx 1.55.</math> (It's actually about <math>1.631.</math>) | ||
+ | |||
+ | So <math>\log_2{6} + \log_3{6}</math> is approximately <math>4.1.</math> The square root of that is just above <math>2,</math> maybe <math>2.02.</math> We cross out all choices below <math>C</math> since they are less than <math>2</math>, and <math>E</math> can't possibly be true unless either <math>\log_2{6}</math> and/or <math>\log_3{6}</math> is <math>0</math> (you can prove this by squaring). Thus, the only feasible answer is $\text{(D).} | ||
== Video Solution == | == Video Solution == |
Revision as of 12:09, 13 August 2021
Contents
- 1 Problem
- 2 Solution 1 (Observations)
- 3 Solution 2 (Properties of Logarithms: Direct)
- 4 Solution 3 (Properties of Logarithms: Stepwise)
- 5 Solution 4 (Change of Base Formula)
- 6 Solution 5 (Solution 1 but more detailed)
- 7 Video Solution
- 8 Video Solution
- 9 Video Solution (Meta-Solving Technique)
- 10 See Also
Problem
Which of the following is the value of
Solution 1 (Observations)
Using the knowledge of the powers of and
we know that
and
Therefore,
Only choices
and
are greater than
but
is certainly incorrect--if we compare the squares of the original expression and
then they are clearly not equal. So, the answer is
~Baolan
~Solasky (first edit on wiki!)
~chrisdiamond10
~MRENTHUSIASM (reformatted and merged the thoughts of all contributors)
Solution 2 (Properties of Logarithms: Direct)
Note that:
We use these properties of logarithms to rewrite the original expression:
~MRENTHUSIASM
Solution 3 (Properties of Logarithms: Stepwise)
. If we call
, then we have
. So our answer is
.
~JHawk0224
Solution 4 (Change of Base Formula)
First,
From here,
Finally,
Answer:
Note that in this solution, even the most minor steps have been written out. In the actual test, this solution would be quite fast, and much of it could easily be done in your head.
~ TheBeast5520
Solution 5 (Solution 1 but more detailed)
Note: Only use this method if all else fails and you cannot find a way to simplify the logarithms.
We can see that is greater than
and less than
. Additionally, since
is halfway between
and
, knowing how exponents increase more the larger
is, we can deduce that
is just above halfway between
and
. We can guesstimate this as
(It's actually about
)
Next, we think of This is greater than
and less than
.
is halfway between
and
and similar to the logic for
we know that
is just above halfway between
and
. We guesstimate this as
(It's actually about
)
So is approximately
The square root of that is just above
maybe
We cross out all choices below
since they are less than
, and
can't possibly be true unless either
and/or
is
(you can prove this by squaring). Thus, the only feasible answer is $\text{(D).}
Video Solution
~IceMatrix
Video Solution
https://youtu.be/RdIIEhsbZKw?t=1463
~ pi_is_3.14
Video Solution (Meta-Solving Technique)
https://youtu.be/GmUWIXXf_uk?t=1298
~ pi_is_3.14
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.