Lesson Solving advanced logarithmic equations
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<H2>Solving advanced logarithmic equations</H2> <H3>Problem 1</H3>Solve an equation {{{(sqrt(x))^(log((x)))}}} = 100. <B>Solution</B> <pre> The original equation is {{{(sqrt(x))^(log((x)))}}} = 100. The domain is { x | x > 0 }. The equation is equivalent to {{{x^((1/2)*log((x)))}}} = 100. Square both sides. You will get {{{x^(log((x)))}}} = 10000. Take logarithm base 10 from both sides. You will get log(x) * log(x) = log(10000), or (log(x))^2 = log(10000), or (log(x))^2 = 4. Take square root from both sides. You will get log(x) = +/- 2. So we have two solutions 1) log(x) = 2, x = {{{10^2}}} = 100, and 2) log((x) = -2, x = {{{10^(-2)}}} = {{{1/100}}}. <U>ANSWER</U>. The original equation has two solutions, x = 100 and x = {{{1/100}}}. </pre> <H3>Problem 2</H3>Solve an equation {{{x^log(10,(x))}}} = 10. Find all its roots. <B>Solution</B> <pre> You are given this equation {{{x^log(10,(x))}}} = 10. Take log(base 10) from both sides. You will get {{{log(10,(x))*log(10,(x))}}} = 1, or {{{(log(10,(x)))^2}}} = 1. Take the square root from both sides {{{log(10,(x))}}} = +/- 1. Case 1. {{{log(10,(x))}}} = 1. It implies x = 10. Case 2. {{{log(10,(x))}}} = -1. It implies x = {{{1/10}}}. <U>ANSWER</U>. The given equation has 2 (two, TWO) solutions x = 10 and x = {{{1/10}}}. </pre> <H3>Problem 3</H3>Solve an equation {{{log(5,(sqrt(x)))}}} = {{{log(sqrt(x),(5))}}} <B>Solution</B> <pre> {{{log(5, (sqrt(x)))}}} = {{{log(sqrt(x), (5))}}}.........change to base {{{10}}} {{{log(sqrt(x))/log((5))}}} = {{{log((5))/ log(sqrt(x))}}}........cross multiply {{{log((sqrt(x)))* log((sqrt(x)))}}} = {{{log((5))*log((5))}}} {{{(log((sqrt(x))))^2}}} = {{{(log((5)))^2}}}........it implies {{{log((sqrt((x))))}}} = +/- {{{log((5))}}} If {{{log((sqrt(x)))}}} = {{{log((5))}}} then sqrt(x) = 5; hence, x = 25. If {{{log((sqrt(x)))}}} = -{{{log((5))}}} then sqrt(x) = 1/5; hence, x = 1/25. <U>ANSWER</U>. There are two solutions, x= 25 and/or x = 1/25. </pre> <H3>Problem 4</H3>Solve this logarithmic-exponential equation {{{2^log(2,(2x + 3))}}} + {{{5^log(5,(x + 7))}}} = {{{7^log(7,(2x + 18))}}}. <B>Solution</B> <pre> Your starting equation is {{{2^log(2,(2x + 3))}}} + {{{5^log(5,(x + 7))}}} = {{{7^log(7,(2x + 18))}}} Due to the basic elementary properties of logarithms, {{{2^log(2,(2x + 3))}}} = 2x + 3 {{{5^log(5,(x + 7))}}} = x + 7 {{{7^log(7,(2x + 18))}}} = 2x + 18. So, the original equation takes the form (2x+3) + (x+7) = 2x+18. Simplify and find x 3x + 10 = 2x + 18 3x - 2x = 18 - 10 x = 8. <U>ANSWER</U>. x = 8. </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/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=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=http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson>Using logarithms to solve real world problems</A>, and - <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-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.
