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Difference between revisions of "1969 AHSME Problems/Problem 25"

(Created page with "== Problem == If it is known that <math>log_2(a)+log_2(b) \ge 6</math>, then the least value that can be taken on by <math>a+b</math> is: <math>\text{(A) } 2\sqrt{6}\quad \text...")
 
m (→‎Solution)
 
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== Problem ==
 
== Problem ==
  
−
If it is known that <math>log_2(a)+log_2(b) \ge 6</math>, then the least value that can be taken on by <math>a+b</math> is:
+
If it is known that <math>\log_2(a)+\log_2(b) \ge 6</math>, then the least value that can be taken on by <math>a+b</math> is:
  
 
<math>\text{(A) } 2\sqrt{6}\quad
 
<math>\text{(A) } 2\sqrt{6}\quad
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
−
<math>\fbox{D}</math>
+
We use the logarithm property of addition:
 +
<cmath>
 +
\begin{align*}
 +
\log_2(a)+\log_2(b) \ge 6 &= \log_2(ab) \ge 6\\
 +
&\Rightarrow 2^{log_2(ab)} \ge 2^6\\
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&= ab \ge 64
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\end{align*}</cmath>
 +
Due to the Quadratic [[Optimization]] or the [[AM-GM Inequality]], the least value is obtained when <math>a = b</math>.
 +
Therefore, <math>a = b = 8 \Rightarrow a + b = \boxed{(D)16}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 16:59, 21 April 2020

Problem

If it is known that $\log_2(a)+\log_2(b) \ge 6$, then the least value that can be taken on by $a+b$ is:

$\text{(A) } 2\sqrt{6}\quad \text{(B) } 6\quad \text{(C) } 8\sqrt{2}\quad \text{(D) } 16\quad \text{(E) none of these}$

Solution

We use the logarithm property of addition: \begin{align*} \log_2(a)+\log_2(b) \ge 6 &= \log_2(ab) \ge 6\\ &\Rightarrow 2^{log_2(ab)} \ge 2^6\\ &= ab \ge 64 \end{align*} Due to the Quadratic Optimization or the AM-GM Inequality, the least value is obtained when $a = b$. Therefore, $a = b = 8 \Rightarrow a + b = \boxed{(D)16}$

See also

1969 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Problem 26
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

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