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Maths: Logarithm equations.


1. Solving logarithm equations.
 
Logarithm equations are equations, where at least one of the equation terms is a logarithm expression. Such equations are part of most intermediate mathematics courses and the equations studied usually have one of the following forms:
    logb(a1x + b1) = logb(a2x + b2)
    logb(a1x + b1) = c
    logb(a1x + b1) = logb(a2x + b2) ± logb(a3x + b3)
    logb(a1x + b1) ± logb(a2x + b2) = c
These equations may have one unique solution, or no solution; some of them may possibly have 2 solutions.
I preview to write a tutorial about solving logarithm equations (as I did for linear equations in 1, 2 and 3 variables). Here, just a brief way how to proceed (if you need more details, please have a look at a maths book or search the Internet):
  1. If the equation contains sums or differences of logarithms, use the logarithm formulae to transform the equation terms into products and quotients.
  2. Use the logarithm's inverse fuction (bx) to get rid of the logarithms. You now have a simple equation, mostly linear, sometimes quadratic, in 1 variable.
  3. Solve the simple equation as you usually do. If it is linear, you get 1 unique solution or no solution. If it is quadratic, you get 1 double or 2 different solutions; if the roots are complex numbers, consider the equation having no solution.
  4. Final very important step: Consider all logarithm terms of the original equation and calculate the function's arguments by replacing x with the value(s) found as solution(s). As logarithms are defined only for strictly positive numbers, a negative or zero argument tells you, that the solution, that you found by solving the simple equation, is NOT a VALID solution of the logarithm equation.
2. Logarithm equations online exercise generator.
 
My online web application is based on my Free Pascal desktop application Equations4, that you may download free of charge in the Free Pascal GUI Applications section of my site. As the desktop program, the online application has three difficulty levels and the option to choose between decimal numbers (evaluated with 3 significant decimal digits) or fractions for the solution(s) to be entered. Note, that these options can't be changed during the test.
Use the following link to start the online application. For details about the Perl source code, see below...
3. Logarithm equations Perl script.
 
Will be ready soon...