Lesson Entertainment problems on logarithms

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Entertainment problems on logarithms


Problem 1

If   S = log(1/2)  +  log(2/3)  +  log(3/4)  +  . . .  +  log(998/999)  +  log(999/1000),

where the base of the logarithm is  10,  what is the integer value of  S  in simplest form?

Solution 

If  S = (log 1/2) + (log 2/3) + (log 3/4) + ... + (log 998/999) + (log 999/1000),


then  S = .


Now, in the global product, cancel the denominators and numerators in all neighbor fractions, and you will get simple expression


    S = log%28%281%2F1000%29%29 = -3.


THEREFORE, the final   A N S W E R   is  S = -3.

Problem 2

If   ln{ln[ln(lnx)]} = 0,  where the base of each natural  log  is  e,  then   x = e%5Ek   for some positive real value of  k.   Find the value of  k.

Solution 

If  ln{ln[ln(lnx)]} = 0,  then

       ln[ln(lnx)]  = 1,  then

          ln(ln(x)) = e, then

             ln(x)  = e%5E%28e%29,  then

                x   = e%5E%28e%5Ee%29.


The problem asks to find such real value "k" that

                x = e%5Ek = e%5E%28e%5Ee%29.


So, this value of "k" is  k = e%5Ee = 2.71828%5E2.71828 = 15.15421   (approximately).


ANSWER.   k = e%5Ee = 2.71828%5E2.71828 = 15.15421   (approximately).

Problem 3

Given that  P > 1  and   1%2Flog%282%2CP%29 + 1%2Flog%283%2CP%29 + 1%2Flog%285%2CP%29 + 1%2Flog%287%2CP%29 + 1%2Flog%2811%2CP%29 = 1%2Flog%28x%2CP%29,
find the integer value of  x.

Solution 

Use an identity

    1%2Flog%28a%2Cb%29 = log%28b%2Ca%29,

which is valid for any positive real "a" and "b".


Then your equation will take the form

    log%28P%2C2%29 + log%28P%2C3%29 + log%28P%2C5%29 + log%28P%2C7%29 + log%28P%2C11%29 = log%28P%2Cx%29.


It is the same as

    log%28P%2C%282%2A3%2A5%2A7%2A11%29%29 = log%28P%2Cx%29,


which you can transform further

    log%28P%2C%282310%29%29 = log%28P%2C%28x%29%29

    x = 2310.


ANSWER.  x = 2310.


My other lessons in this site on logarithms,  logarithmic equations and relevant word problems are
    - WHAT IS the logarithm,
    - Properties of the logarithm,
    - Change of Base Formula for logarithms,
    - Evaluate logarithms without using a calculator
    - Simplifying expressions with logarithms
    - Solving logarithmic equations,
    - Solving advanced logarithmic equations
    - Solving really interesting and educative problem on logarithmic equation containing a HUGE underwater stone
    - Proving equalities with logarithms
    - Solving logarithmic inequalities
    - Using logarithms to solve real world problems
    - Solving problem on Newton Law of cooling
    - Population growth problems
    - Radioactive decay problems
    - Carbon dating problems
    - Bacteria growth problems
    - A medication decay in a human's body
    - Problems on appreciated/depreciated values
    - Inflation and Salary problems
    - Miscellaneous problems on exponential growth/decay
    - Problems on discretely compound accounts
    - Problems on continuously compound accounts
    - Tricky problem on solving a logarithmic system of equations
    - Entertainment problem: Uninterrupted withdrawing money from a retirement fund
    - Entertainment problems on exponential growth
    - Upper level problems on solving logarithmic equations
    - OVERVIEW of lessons on logarithms, logarithmic equations and relevant word problems

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