Difference between revisions of "1953 AHSME Problems/Problem 22"
(Created page with "This essentially simplifies to log base 3 of (3^3)(3^1/2)(3^2/3) Simplifying the equation, we get log base 3 of 3^3+1/2+2/3 Further simplifying, we get log base 3 of 3^25/6...") |
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− | + | ==Problem== | |
+ | The logarithm of <math>27\sqrt[4]{9}\sqrt[3]{9}</math> to the base <math>3</math> is: | ||
− | + | <math>\textbf{(A)}\ 8\frac{1}{2} \qquad | |
+ | \textbf{(B)}\ 4\frac{1}{6} \qquad | ||
+ | \textbf{(C)}\ 5 \qquad | ||
+ | \textbf{(D)}\ 3 \qquad | ||
+ | \textbf{(E)}\ \text{none of these} </math> | ||
− | + | ==Solution== | |
+ | <math>27\sqrt[4]{9}\sqrt[3]{9}</math> can be rewritten as <math>3^3\cdot 3^\frac{1}{2}\cdot 3^\frac{2}{3}</math>. Using exponent rules, this simplifies to <math>3^\frac{25}{6}</math>. | ||
+ | The problem wants us to find <math>\log_3{27\sqrt[4]{9}\sqrt[3]{9}}</math>. We just found that this is equal to <math>\log_3{3^\frac{25}{6}}</math>. Using logarithm rules, this is equal to <math>\frac{25}{6}\log_3{3}</math>, which is simply <math>\frac{25}{6}</math>. The answer is <math>\boxed{\text{(B)}\ 4\frac{1}{6}}</math>. | ||
− | + | ==See Also== | |
− | + | {{AHSME 50p box|year=1953|num-b=21|num-a=23}} | |
+ | |||
+ | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 20:30, 27 November 2019
Problem
The logarithm of to the base
is:
Solution
can be rewritten as
. Using exponent rules, this simplifies to
.
The problem wants us to find
. We just found that this is equal to
. Using logarithm rules, this is equal to
, which is simply
. The answer is
.
See Also
1953 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
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