Lesson Entertainment problems on logarithms
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<H2>Entertainment problems on logarithms</H2> <H3>Problem 1</H3>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? <B>Solution</B> <pre> If S = (log 1/2) + (log 2/3) + (log 3/4) + ... + (log 998/999) + (log 999/1000), then S = {{{log (((1/2)*(2/3)*(3/4)*ellipsis*(998/999)*(999/1000)))}}}. Now, in the global product, cancel the denominators and numerators in all neighbor fractions, and you will get simple expression S = {{{log((1/1000))}}} = -3. THEREFORE, the final <U>A N S W E R</U> is S = -3. </pre> <H3>Problem 2</H3>If ln{ln[ln(lnx)]} = 0, where the base of each natural log is e, then x = {{{e^k}}} for some positive real value of k. Find the value of k. <B>Solution</B> <pre> If ln{ln[ln(lnx)]} = 0, then ln[ln(lnx)] = 1, then ln(ln(x)) = e, then ln(x) = {{{e^(e)}}}, then x = {{{e^(e^e)}}}. The problem asks to find such real value "k" that x = {{{e^k}}} = {{{e^(e^e)}}}. So, this value of "k" is k = {{{e^e}}} = {{{2.71828^2.71828}}} = 15.15421 (approximately). <U>ANSWER</U>. k = {{{e^e}}} = {{{2.71828^2.71828}}} = 15.15421 (approximately). </pre> <H3>Problem 3</H3>Given that P > 1 and {{{1/log(2,P)}}} + {{{1/log(3,P)}}} + {{{1/log(5,P)}}} + {{{1/log(7,P)}}} + {{{1/log(11,P)}}} = {{{1/log(x,P)}}}, find the integer value of x. <B>Solution</B> <pre> Use an identity {{{1/log(a,b)}}} = {{{log(b,a)}}}, which is valid for any positive real "a" and "b". Then your equation will take the form {{{log(P,2)}}} + {{{log(P,3)}}} + {{{log(P,5)}}} + {{{log(P,7)}}} + {{{log(P,11)}}} = {{{log(P,x)}}}. It is the same as {{{log(P,(2*3*5*7*11))}}} = {{{log(P,x)}}}, which you can transform further {{{log(P,(2310))}}} = {{{log(P,(x))}}} x = 2310. <U>ANSWER</U>. x = 2310. </pre> My other lessons in this site on logarithms, logarithmic equations and relevant word problems are - <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A>, - <A HREF=http://www.algebra.com/algebra/homework/logarithm/Properties-of-the-logarithm.lesson>Properties of the logarithm</A>, - <A HREF=http://www.algebra.com/algebra/homework/logarithm/change-of-base-formula-for-logarithms.lesson>Change of Base Formula for logarithms</A>, - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Evaluate-logarithms-without-using-a-calculator.lesson>Evaluate logarithms without using a calculator</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Simplifying-expressions-with-logarithms.lesson>Simplifying expressions with logarithms</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/How-to-solve-logarithmic-equations.lesson>Solving logarithmic equations</A>, - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-advanced-logarithmi%D1%81-equations.lesson>Solving advanced logarithmic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-really-interesting-and-educative-problem-on-logarithmic-equation-containing-a-HUGE-underwater-stone.lesson>Solving really interesting and educative problem on logarithmic equation containing a HUGE underwater stone</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Proving-equalities-with-logarithms.lesson>Proving equalities with logarithms</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-logarithmic-inequalities.lesson>Solving logarithmic inequalities</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson>Using logarithms to solve real world problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-problem-on-Newton-Law-of-cooling.lesson>Solving problem on Newton Law of cooling</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Population-growth-problems.lesson>Population growth problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Radioactive-decay-problems.lesson>Radioactive decay problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Carbon-dating-problems.lesson>Carbon dating problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Bacteria-growth-problems.lesson>Bacteria growth problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/A-medication-decay-in-a-human%27s-body.lesson>A medication decay in a human's body</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-appreciated-depreciated-values.lesson>Problems on appreciated/depreciated values</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Inflation-and-Salary-problems.lesson>Inflation and Salary problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Miscellaneous-problems-on-exponential-growth-decay.lesson>Miscellaneous problems on exponential growth/decay</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-discretely-compound-accounts.lesson>Problems on discretely compound accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-continuously-compound-accounts.lesson>Problems on continuously compound accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Tricky-problem-on-solving-a-logarithmic-system-of-equations.lesson>Tricky problem on solving a logarithmic system of equations</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Uninterrupted-withdrawing-money-from-a-retirement-fund.lesson>Entertainment problem: Uninterrupted withdrawing money from a retirement fund</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Entertainment-problems-on-exponential-growth.lesson>Entertainment problems on exponential growth</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Upper-level-problem-on-solving-logarithmic-equations.lesson>Upper level problems on solving logarithmic equations</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/OVERVIEW-of-lessons-on-logarithms-logarithmic-eqns-and-relevant-word-probs.lesson>OVERVIEW of lessons on logarithms, logarithmic equations and relevant word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
