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):
- If the equation contains sums or differences of logarithms, use the logarithm formulae to transform the equation terms into products and quotients.
- 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.
- 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.
- 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.