Properties of Logarithms
Properties of Logarithmic Functions
Let b>0, b≠1, let x and y be positive numbers, and let r be any real number. Then the following properties hold:
The range of logbx is all real numbers.
The domain of logbx is all positive real numbers.
For b>1, logbx>0 for x>1 and logbx<0 for 0<x<1; for 0<b<1 the inequalities reverse.
If x>y and b>1 then logbx>logby. If x>y and 0<b<1 then logbx<logby. That is, logbx is an increasing function if b>1 and a decreasing function if 0<b<1.
x=logby exactly when y=bx. That is, the logarithmic and exponential functions with the same base are inverses of each other. In particular, logbbx=x=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?
The "log" button represents the base 10 logarithmic function. The calculator displays answers to 2 decimal places.
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