Lesson Solving advanced logarithmic equations

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> Lesson Solving advanced logarithmic equations      Log On


   

This Lesson (Solving advanced logarithmic equations) was created by by ikleyn(52776) About Me : View Source, Show
About ikleyn:

Solving advanced logarithmic equations


Problem 1

Solve an equation   %28sqrt%28x%29%29%5E%28log%28%28x%29%29%29 = 100.

Solution 

The original equation is

    %28sqrt%28x%29%29%5E%28log%28%28x%29%29%29 = 100.


The domain is  { x | x > 0 }.


The equation is equivalent to

    x%5E%28%281%2F2%29%2Alog%28%28x%29%29%29 = 100.


Square both sides.   You will get

    x%5E%28log%28%28x%29%29%29 = 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%5E2 = 100,   and


    2)  log((x) = -2,  x = 10%5E%28-2%29 = 1%2F100.


ANSWER.  The original equation has two solutions,  x = 100  and  x = 1%2F100.

Problem 2

Solve an equation   x%5Elog%2810%2C%28x%29%29 = 10.   Find all its roots.

Solution 

You are given this equation

    x%5Elog%2810%2C%28x%29%29 = 10.


Take log(base 10) from both sides.  You will get

    log%2810%2C%28x%29%29%2Alog%2810%2C%28x%29%29 = 1,

or

    %28log%2810%2C%28x%29%29%29%5E2 = 1.


Take the square root from both sides

    log%2810%2C%28x%29%29 = +/- 1.


Case 1.   log%2810%2C%28x%29%29 = 1.

          It implies  x = 10.


Case 2.  log%2810%2C%28x%29%29 = -1.

         It implies  x = 1%2F10.


ANSWER.  The given equation has 2 (two, TWO) solutions  x = 10  and  x = 1%2F10.

Problem 3

Solve an equation   log%285%2C%28sqrt%28x%29%29%29 = log%28sqrt%28x%29%2C%285%29%29

Solution 

log%285%2C+%28sqrt%28x%29%29%29 = log%28sqrt%28x%29%2C+%285%29%29.........change to base 10 

log%28sqrt%28x%29%29%2Flog%28%285%29%29 = log%28%285%29%29%2F+log%28sqrt%28x%29%29........cross multiply

log%28%28sqrt%28x%29%29%29%2A+log%28%28sqrt%28x%29%29%29 = log%28%285%29%29%2Alog%28%285%29%29

%28log%28%28sqrt%28x%29%29%29%29%5E2 = %28log%28%285%29%29%29%5E2........it implies

log%28%28sqrt%28%28x%29%29%29%29 =  +/- log%28%285%29%29


If log%28%28sqrt%28x%29%29%29 =  log%28%285%29%29   then  sqrt(x) =  5;  hence,  x = 25.

If log%28%28sqrt%28x%29%29%29 =  -log%28%285%29%29  then  sqrt(x) =  1/5;  hence,  x = 1/25.


ANSWER.  There are two solutions,  x= 25  and/or  x = 1/25.

Problem 4

Solve this logarithmic-exponential equation   2%5Elog%282%2C%282x+%2B+3%29%29 + 5%5Elog%285%2C%28x+%2B+7%29%29 = 7%5Elog%287%2C%282x+%2B+18%29%29.

Solution 

Your starting equation is

    2%5Elog%282%2C%282x+%2B+3%29%29 + 5%5Elog%285%2C%28x+%2B+7%29%29 = 7%5Elog%287%2C%282x+%2B+18%29%29


Due to the basic elementary properties of logarithms,

    2%5Elog%282%2C%282x+%2B+3%29%29 = 2x + 3

    5%5Elog%285%2C%28x+%2B+7%29%29 = x + 7

    7%5Elog%287%2C%282x+%2B+18%29%29 = 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.


ANSWER.  x = 8.


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,
    - Proving equalities with logarithms
    - Solving logarithmic inequalities
    - Solving really interesting and educative problem on logarithmic equation containing a HUGE underwater stone
    - Using logarithms to solve real world problems,  and
    - 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 logarithms
    - Entertainment problems on exponential growth
    - Upper level problems on solving logarithmic equations
    - OVERVIEW of lessons on logarithms, logarithmic equations and relevant word problems

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

This lesson has been accessed 2022 times.