Lesson Tricky problem on solving a logarithmic system of equations
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<H2>Tricky problem on solving a logarithmic system of equations</H2> <H3>Problem 1</H3>For given system of equations {{{ log (a,x) = 5 }}} {{{ y = log (a,y) }}} exclude the parameter " a " and express x as a function of y. <B>Solution</B> <pre> You are given two equations (system of two equations) {{{log(a,(x))}}} = 5 (1) {{{log(a,(y))}}} = y (2) Divide equation (1) by equation (2). You will get {{{log(a,(x))/log(a,(y))}}} = {{{5/y}}}. (3) Apply the change-of-base formula for logarithms to the left side of equation (3). Instead of equation (3), you will get then {{{log(y,(x))}}} = {{{5/y}}}. (4) It means that x = y^(5/y). (5) It is the sough expression of x via y, with parameter "a" excluded. Expression (5) is your answer. It allows calculate the value of x for any given positive value of y. </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/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-logarithms.lesson>Entertainment problems on logarithms</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.
