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Properties of Logarithms
Logarithmic functions and exponential functions are connected to one another in that they are inverses of each other. You may recall that when two functions are inverses of each other, the x and y coordinates are swapped. This leads to the most basic property involving logarithms which allows you to move back and forth between logarithmic and exponential forms of an expression.
is exactly the same as
You can use the above property to change a logarithmic expression into an exponential expression or an exponential expression into a logarithmic expression.

Let's Practice:
  1. Change the exponential equation to logarithmic form.
When changing between logarithmic and exponential forms, the base is always the same. In the exponential form in this problem, the base is 2, so it will become the base in our logarithmic form.

Because logarithms and exponents are inverses of each other, the x and y values change places. It doesn’t matter if x and y are variables or numbers. In this case the 2 is “attached” to the 3 and the 8 is by itself. So when we switch those values, the 2 is by itself and the 8 is “attached”

So written is logarithmic form is
  1. Change into exponential form.
Since the base is the same whether we are dealing with an exponential or a logarithm, the base for this problem will be 5.

We will exchange the 4 and the 625. The 625 was attached to the 5 and the 4 was by itself. In the logarithmic form, the 625 will be by itself and the 4 will be attached to the 5.

So in exponential form is .

This problem is nice because you can check it on your calculator to make sure your exponential equation is correct.

In addition to the property that allows you to go back and forth between logarithms and exponents, there are other properties that allow you work with logarithmic expressions.
Properties of Logarithms:
  1. loga1 = 0
You can verify why this works by changing to an exponential form and getting and anything to the zero power is 1. This property says that no matter what the base is, if you are taking the logarithm of 1, then the answer will always be 0.
You can verify this by changing to an exponential form and getting . This property says that if the base and the number you are taking the logarithm of are the same, then your answer will always be 1.
This property allows you to take a logarithmic expression of two things that are multiplied, then you can separate those into two distinct expressions that are added together. You can also go the other way. Two log expressions that are added can be combined into a single log expression using multiplication.
This property allows you to take a logarithmic expression involving two things that are divided, then you can separate those into two distinct expressions that are subtracted. You can also go the other way. Two log expressions that are subtracted can be combined into a single log expression using division.
This property will be very useful in solving equations and application problems. It allows you to take the exponent in a logarithmic expression and bring it to the front as a coefficient. You can also go the other way and move a coefficient up so that it becomes an exponent.
So if properties 3, 4 and 5 can be used both ways, how do you know what should be done? That depends on the type of problem that is being asked. If you are being asked to combine log expressions into a single expression, you’ll want to use the property from right to left. But if you are trying to break up a single log expression into its separate parts, you’ll want to use the property from left to right.

Let's Practice:
  1. Use the properties of logs to write as a single logarithmic expression.
Since this problem is asking us to combine log expressions into a single expression, we will be using the properties from right to left.

We usually begin these types of problems by taking any coefficients and writing them as exponents.
Now there are two log terms that are added. We can combine those into a single log expression by multiplying the two parts together.
We have now condensed the original problem into a single logarithmic expression.
  1. Expand the expression .
Since we are trying to break the original expression up into separate pieces, we will be using our properties from left to right.

We begin by taking the three things that are multiplied together and separating those into individual logarithms that are added together.
There is an exponent in the middle term which can be brought down as a coefficient. This gives us
There are no terms multiplied or divided nor are there any exponents in any of the terms. We have expanded this expression as much as possible.

Examples
Example Write as a single log expression.
What is your answer?
 
Example Write as a single log expression.
What is your answer?
 
Example Write as a single log expression.
What is your answer?
 
Example Expand .
What is your answer?
 
Example Expand .
What is your answer?
 
Example Expand .
What is your answer?
 

While it may seem silly to break apart or condense logarithmic expressions and then stop, these skills are essential when solving logarithmic equations. You must be able to work the problems in this lesson in order be successful when solving equations.


S Taylor

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