Remember that *log _{a}M =x *means
exactly the same thing as

This is the key to solving equations in which logarithms appear.

For example: Suppose we want to
solve the equation *log _{2} y = 3. * Since this means
exactly the same thing as

Here's a slightly harder problem: Solve the
equation *log _{2} (5z+ 1) = 4. *Since this means exactly
the same thing as

**Exercises I**

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
*log _{2} (x + 1)
+log_{2} (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

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.

**Exercises II**

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!

**Exercises III**

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.*