Remember that logaM =x means exactly the same thing as ax = M , that is,
This is the key to solving equations in which logarithms appear.
For example: Suppose we want to solve the equation log2 y = 3. Since this means exactly the same thing as 23 = y, the equation pretty much solves itself!!
Here's a slightly harder problem: Solve the equation log2 (5z+ 1) = 4. Since this means exactly the same thing as 24 = 5z+1 , we need only solve the ordinary equation 16 = 5z +1 . So the solution is z = 3.
Solve the following.
You should see from these examples that the basic strategy is to convert the equation involving logarithms to one that doesn't involve logs (by using the equivalent exponential form of the equation), and then solving the converted equation. In order to do this you will sometimes need to combine the logarithmic terms.
Example: Solve log2 (x + 1) +log2 (x) = 1.
The first step is to use properties of logarithms to combine the logarithmic terms. Using product rule we get:
which is the same as
This is a quadratic equation, and you can easily solve it. The solutions of this last equation are
BUT NOTE!! ONLY x = 1 is a solution of the original equation! x = -2 cannot be a solution, because you can't take logs of negative numbers, so if you try to put x = -2 into the original logarithmic equation you would get
Neither logarithm makes sense, so -2 can't be a solution.
This is the tricky part of solving logarithmic equations: noticing and eliminating those pesky numbers that appear when you try to solve a given equation, which are not solutions. Remember, it is perfectly possible for a logarithmic equation not to have any solutions.
Determine the number of solutions for each of the following logarithmic equations. (Just the number of solutions, not their actual values.) You might have to use the quadratic formula for some of these. You will also have to be observant!
Each of the following has exactly one solution. Find it. Remember, your converted equation may have a solution that is not a solution of the original equation.
Precalculus Tutorials, B. Kaskosz and L. Pakula, 2002.