Properties of Logarithms - Maple Help

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Properties of Logarithms

Properties of Logarithmic Functions

Let b>0, b1, let x and y be positive numbers, and let r be any real number. Then the following properties hold:

 

1. 

The range of logbx is all real numbers.

2. 

The domain of logbx is all positive real numbers.

3. 

For b&gt;1, logbx&gt;0 for x&gt;1 and logbx<0 for 0<x<1; for 0<b<1 the inequalities reverse.

4. 

logb1&equals;0

5. 

logbxy&equals;logbx&plus;logby

6. 

logbxy&equals;logbxlogby

7. 

logb1x&equals;logbx

8. 

logbxr&equals;r logbx

9. 

If x&gt;y and b&gt;1 then logbx&gt;logby. If x&gt;y and 0<b<1 then logbx<logby. That is, logbx is an increasing function if b&gt;1 and a decreasing function if 0<b<1.

10. 

x&equals;logby exactly when y&equals;bx. That is, the logarithmic and exponential functions with the same base are inverses of each other. In particular, logbbx&equals;x&equals;blogbx.

Using the properties of logarithms

The calculator shown here is missing a few keys (no multiplication or division keys). Nonetheless, it is still possible to perform any arithmetic calculation involving only +, -, ×, or ÷ operations. This is because the calculator has "10 to the power of" and "logarithm base 10" keys.

 

Try it out. Can you compute these values?

• 

4/3

• 

93123×23485

• 

5217.308&plus;234.33×941.226177.332

The "log" button represents the base 10 logarithmic function. The calculator displays answers to 2 decimal places.

 

    

   

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