Logarithmic Equation Solving

Recall the basic rules from your practice session. We'll need them throughout this section. Notice that we changed our perspective in the equalities.

The Key Principle x = logb y is the same as y = b x Properties of logarithms log x + log y = log(xy) log x - log y = log(x/y) n log x = log x n log x = log y means x = y (Change of Base Formula)

We need to solve logarithmic equations occasionally. Sometimes we do it graphically, but throughout this course be prepared to solve them algebraically.

Many logarithmic problems eventually look like . When we can reduce to this form, the solving just requires us to rewrite the expression to an exponential . In a sense when we can do this, we turn the log problem into a simple exponential evaluation process.

Here are some typical examples:

Example 1:

In the problem to the right you many of the elements of solving a logarithmic equation.

1. Isolate the logarithmic term
2. Eliminate the coefficient by multiplication or division.
3. Rewrite the expression to an exponential form. Here base 10 is required.
4. Isolate the variable using normal algebraic processes.

Example 2:

In the problem to the right you have another element of solving a logarithmic equation.

1. Isolate the logarithmic term, in this case by subtraction
2. Eliminate the coefficient by multiplication or division.
3. Rewrite the expression to an exponential form. Here base e is required.
4. Isolate the variable using normal algebraic processes.

Example 3:

In the problem to the right we are close to having all elements of solving a logarithmic equation.

1. Isolate the logarithmic term, in this case we subtract.
2. Eliminate the coefficient by multiplication.
3. Rewrite the expression to an exponential form. Here base e is required.
4. Isolate the variable using normal algebraic processes.

Example 4:

Now we're getting serious. In the problem to the right, notice we handle as much solving as possible before we rewrite to the exponential form.

1. Isolate the logarithmic term, in this case we subtract.
2. Eliminate the coefficient by division.
3. Rewrite the expression to an exponential form. Here base 10 is required.
4. Isolate the variable using normal algebraic processes.

Example 5:

No more simplest stuff. Let's start using the properties of logarithms.

1. Combine all logs into a single using properties 1, 2, and 3.
2. Rewrite the expression to an exponential form. Here base 2 is required.
3. Solve for the variable using normal algebraic processes.
4. Now comes the ugly part: Every potential solution must be checked against the original expression. This was true in examples 1 - 4, but there was no problem with any of those solutions.
   • Notice that x = 2 checks fine.
   • Notice that x = 0 is impossible since is defined for every base.
5. So the only correct solution is x = 2 .
6. Also notice, without using the change of base formula on the two log expression, you cannot solve this problem graphically!

Example 6:

Another example using the properties of logarithms.

1. Sometimes we can use property 4.
• But this does not mean that we cancelled the word log.
  This symbol represents a function just like f(x).
• Should you ever do this nonsense, , as a part of solving, you may be certain that points will be deducted!
• Consider that if this is legal, so is this process:. Check both sides through your calculator and you'll see that it doesn't work.
2. Solve for the variable using normal algebraic processes.
3. The solution checks easily.

Now for some practical applications:

Here's the bad news: Most relationships that work logarithmically are either noisy or disastrous! You see, the most common examples are in the power related to earthquakes, tornados and sound levels.

You should have heard of the scale used to measure the power of earthquakes. It is called the Richter Scale (after the person who developed it) and it works in the logarithmic scale.

The equation is below. M is the magnitude of the earthquake and S is the standard earthquake which is a wimpy ground movement of 10^-4cm.

Suppose an earthquake occurred at the Grand Canyon with a magnitude of 3.5 on the Richter scale. Someone from San Francisco snickered when they heard about it since they had survived the Loma Prieta earthquake of 1989. It was almost 4000 times more intense. What was the magnitude of the Loma Prieta earthquake?

1. Let L and G represent the magnitudes of the Loma Prieta and Grand Canyon events, respectively.
2. We have the two equations to the right.
3. We sue substation, then apply the properties of logs.
4. Note also, the properties were used only as far as needed to complete the solution.