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Difference between revisions of "1967 AHSME Problems/Problem 23"

(Created page with "== Problem == If <math>x</math> is real and positive and grows beyond all bounds, then <math>\log_3{(6x-5)}-\log_3{(2x+1)}</math> approaches: <math>\textbf{(A)}\ 0\qquad \textbf...")
 
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== Solution ==
 
== Solution ==
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<math>\fbox{B}</math>
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Since <math>\log_b x - \log_b y = \log_b \frac{x}{y}</math>, the expression is equal to <math>\log_3 \frac{6x - 5}{2x + 1}</math>.
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The expression <math>\frac{6x - 5}{2x + 1}</math> is equal to <math>3 - \frac{8}{2x + 1}</math>.  As <math>x</math> gets large, the second term approaches <math>0</math>, and thus <math>\frac{6x - 5}{2x + 1}</math> approaches <math>3</math>.  Thus, the expression approaches <math>\log_3 3</math>, which is <math>1</math>.
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Alternately, we divide the numerator and denominator of <math>\frac{6x - 5}{2x + 1}</math> by <math>x</math> to get <math>\frac{6 - \frac{5}{x}}{2 + \frac{1}{x}}</math>.  As <math>x</math> grows large, both fractions approach <math>0</math>, leaving <math>\frac{6}{2} = 3</math>, and so the expression approaches <math>\log_3 3 = 1</math>.
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With either reasoning, the answer is <math>\fbox{B}</math>
  
 
== See also ==
 
== See also ==
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{{AHSME box|year=1967|num-b=22|num-a=24}}   
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{{AHSME 40p box|year=1967|num-b=22|num-a=24}}   
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:40, 16 August 2023

Problem

If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{no finite number}$

Solution

Since $\log_b x - \log_b y = \log_b \frac{x}{y}$, the expression is equal to $\log_3 \frac{6x - 5}{2x + 1}$.

The expression $\frac{6x - 5}{2x + 1}$ is equal to $3 - \frac{8}{2x + 1}$. As $x$ gets large, the second term approaches $0$, and thus $\frac{6x - 5}{2x + 1}$ approaches $3$. Thus, the expression approaches $\log_3 3$, which is $1$.

Alternately, we divide the numerator and denominator of $\frac{6x - 5}{2x + 1}$ by $x$ to get $\frac{6 - \frac{5}{x}}{2 + \frac{1}{x}}$. As $x$ grows large, both fractions approach $0$, leaving $\frac{6}{2} = 3$, and so the expression approaches $\log_3 3 = 1$.

With either reasoning, the answer is $\fbox{B}$

See also

1967 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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
All AHSME Problems and Solutions

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