Why is b>1 and x,y positive in logarithm definition?

In summary, the conversation discusses the definition of logarithmic functions and natural logarithms. The formula for logarithms is given as log_b(xy) = log_b(x) + log_b(y), with the condition that b>1 and b is not equal to zero, and that x and y are positive numbers. The condition is necessary because without it, the function would not be well-defined for negative numbers or zero. The conversation also mentions that the function is continuous and monotonically increasing or decreasing in certain intervals, depending on the value of the base. Overall, the conversation highlights the importance of understanding the conditions and limitations of logarithmic functions.
  • #1
UchihaClan13
145
12
A simple doubt came to my mind while browsing through logarithmic functions and natural logarithms
we define
$$\log_b(xy) = \log_b(x) + \log_b(y)$$
Here
why is the condition imposed that b>1 and b is not equal to zero and that x and y are positive numbers?
Is it something to do with the function being continuous and monotonically increasing or decreasing in certain intervals(1,infinity) and (0,1) respectively?UchihaClan13
 
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  • #2
I fixed the formula, the image didn't get displayed.

You need the three terms to be defined to have an equation. Unless you introduce complex numbers, the logarithm is not defined for negative numbers, and a zero or negative base doesn't make sense, and b=1 doesn't work either. A base between 0 and 1 would be possible, but odd.
 
  • #3
X and Y must be positive because if logA(X) = B, then AB=X. Since you cannot raise A to any power and get a negative number (except possibly with complex numbers, not sure) X must be positive. The same applies for Y.
 
  • #4
For ##b < 1## one gets ##\log_b x = - \log_{\frac{1}{b}} x## and end up with a basis above ##1##.
Thus there is simply no need to consider basis below ##1##. And of course ##b=1## cannot be defined at all.
 
  • #5
UchihaClan13 said:
A simple doubt came to my mind while browsing through logarithmic functions and natural logarithms
we define
$$\log_b(xy) = \log_b(x) + \log_b(y)$$

UchihaClan13

This is not a definition.
 
  • #6
Math_QED said:
This is not a definition.

It can be.
 
  • #7
Thats just definition of logarithms
 

FAQ: Why is b>1 and x,y positive in logarithm definition?

1. What is the sign convention for logarithms?

The sign convention for logarithms states that the logarithm of a number greater than 1 will be positive, while the logarithm of a number between 0 and 1 will be negative. This convention is based on the properties of logarithms and makes calculations easier.

2. Why is it important to follow the sign convention for logarithms?

Following the sign convention for logarithms ensures consistency in calculations and avoids confusion. It also helps in determining the direction of change when using logarithms in equations.

3. Can the sign convention for logarithms be applied to complex numbers?

No, the sign convention for logarithms only applies to real numbers. Complex numbers have a different convention for their logarithms, which involves converting them into polar form.

4. How does the sign convention for logarithms affect the graph of a logarithmic function?

The sign convention for logarithms affects the graph of a logarithmic function by determining the behavior of the function for different inputs. For inputs greater than 1, the function will have a positive slope and for inputs between 0 and 1, the function will have a negative slope.

5. Are there any exceptions to the sign convention for logarithms?

Yes, there are some exceptions to the sign convention for logarithms. For example, when taking the logarithm of a negative number, the result will be a complex number with a negative real part. Also, when using logarithms in complex analysis, a different convention may be used.

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