GATE Overflow | Mock GATE | Test 1 | Question: 7

Question

What is the value of $\{ ( 1/ \log_3 60)+ (1/ \log_4 60 ) + (1/ \log_5 60) \}$?

• A. $0$
  
• B. $1$
  
• C. $5$
  
• D. $60$

Comments

B )

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To enter $log_360$ in GATE virtual calculator, first enter 60. Then click $log_yx$ then enter 3.

In other words, to use $log_yx$ first enter x, then press $log_yx$, then enter y.

 

https://www.tcsion.com/OnlineAssessment/ScientificCalculator/Calculator.html#nogo

Answer 1

we know that $\log_b a = 1/ log_a b$

So $1/log_3\ 60 + 1/log_4\ 60 + 1/log_4\ 60$

$= log_{60}\ 3 + log_{60}\ 4 + log_{60}\ 5$

$= log_{60}\ 3*4*5$              ($\because log_d\ a + log_d\ b + log_d\ c = log_d\ a*b*c$)

$= log_{60}\ 60$

$= 1$

Answer 2

$\frac{1}{log_{3}60}+\frac{1}{log_{4}60}+\frac{1}{log_{5}60}$

=$\frac{1}{log_{3}3+log_{3}4+log_{3}5}+\frac{1}{log_{4}3+log_{4}4+log_{4}5}+\frac{1}{log_{5}3+log_{5}4+log_{5}5}$

=$\frac{1}{1+log_{3}4+log_{3}5}+\frac{1}{log_{4}3+1+log_{4}5}+\frac{1}{log_{5}3+log_{5}4+1}$

=$\frac{1}{1+\frac{log4}{log3}+\frac{log5}{log3}}+\frac{1}{\frac{log3}{log4}+1+\frac{log5}{log4}}+\frac{1}{\frac{log3}{log5}+\frac{log4}{log5}+1}$

=$\frac{log3}{log3+log4+log5}+\frac{log4}{log3+log4+log5}+\frac{log5}{log3+log4+log5}$

=$\frac{log3+log4+log5}{log3+log4+log5}$

=$1$

Option B.

Comments

No need to solve this much.Just use Virtual Calc.
 

$log_yx$ followed by $\frac{1}{x}$ followed by $M+$