Present Year with Logs.

Algebra Level pending

Let $a$ , $b$ , $c$ , $d$ , $e$ , and $f$ be integers satisfying:

$\color{#3D99F6}a\log_{10} 2+b\log_{10} 3+c\log_{10} 5+d\log_{10} 7+e\log_{10} 11+f\log_{10} 13 = 2018$

Then what is $\color{#20A900}a+b+c+d+e+f =?$

The answer is 4036.

1 solution

Hana Wehbi
Jul 30, 2018

$\color{#3D99F6}a\log_{10} 2+b\log_{10} 3+c\log_{10} 5+d\log_{10} 7+e\log_{10} 11+f\log_{10} 13 = 2018\implies$

$2^a3^b5^c7^d11^e13^f=10^{2018}\implies a=c=2018 \text { and } b= d=e = f=0\implies$

$a+b+c+d+e+f=2018+2018=4036$

Just wondering why you have to place "and" in LaTex. You can do \ (a\ ), \ (b\ ), and \ (c\ ).

Chew-Seong Cheong - 2 years, 10 months ago

I feel it is easier since it is in the same brackets. I used to do your method long time. Sir, if you have any advice, l would be glad to consider it.

Hana Wehbi - 2 years, 10 months ago

We can actually do \ (a, b, c, d, e,\ ) and \ (f\ ). Then it saves \text{ and } and the "and" is not in roman style. But I am very particular of presentation. The commas in LaTex is different in size from those outside.

Chew-Seong Cheong - 2 years, 10 months ago

@Chew-Seong Cheong – Thank you, it makes sense.

Hana Wehbi - 2 years, 10 months ago

Present Year with Logs.

The answer is 4036.

1 solution

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