Past, present and future

Algebra Level 2

Is the following equation true?

ln 2014 log 2014 + log 2015 ln 2015 = log 2016 ln 2016 + ln 2017 log 2017 \large\dfrac{\ln 2014}{\log 2014}+\dfrac{\log 2015}{\ln 2015}= \dfrac{\log 2016}{\ln 2016}+ \dfrac{\ln 2017}{\log 2017}

Here, log x \log x denotes log 10 x . \log_{10} x.

No Yes There is insufficient information

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1 solution

Akshat Sharda
Dec 31, 2015

Let,

n = ln 2014 log 2014 + log 2015 ln 2015 = log 2014 log e log 2014 + log 2015 log 2015 log e = 1 log e + log e \begin{aligned} n & = \frac{\text{ln }2014}{\log 2014 }+\frac{\log 2015 }{\text{ln }2015} \\ & = \frac{\frac{\log 2014 }{\log e}}{\log 2014 }+\frac{\log 2015 }{\frac{\log 2015 }{\log e }} \\ & = \frac{1}{\log e}+\log e \end{aligned}

And,

m = ln 2017 log 2017 + log 2016 ln 2016 = log 2017 log e log 2017 + log 2016 log 2016 log e = 1 log e + log e \begin{aligned} m & = \frac{\text{ln }2017}{\log 2017 }+\frac{\log 2016 }{\text{ln }2016} \\ & = \frac{\frac{\log 2017 }{\log e}}{\log 2017 }+\frac{\log 2016 }{\frac{\log 2016 }{\log e }} \\ & = \frac{1}{\log e}+\log e \end{aligned}

Now, we can see that n = m n=m .

actually since log(x) means ln(x), both sides are 2.

Aareyan Manzoor - 5 years, 5 months ago

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I think you're probably right. I assumed log(x) was equal to log base 10 of x, because that's what it means in the US. It's cool that it works either way, though.

Chris Callahan - 5 years, 5 months ago

I think most people use base-10 when they write log(x). ln(x) is base-e.

Anupam Nayak - 5 years, 5 months ago

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well. base-e in calculus we dont even use base-10. see the wikipedia page on taylor series. you will find queries about natural log as log(n). also in wolfram alpha log is used as base e.

Aareyan Manzoor - 5 years, 5 months ago

ln x is not equal to log x/log e because the base in log x is e so it is ln x only and log e is 1. ln x is equal to 2.303 times log to base 10 of x.

rushi panmand - 5 years, 5 months ago

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