Question

Which properties are present in a table that represents a logarithmic function in the form y=logb^x when b>1

Answer 1

The properties present in a table representing a logarithmic function in the form y=logb^x when b>1 are:

1. The values of x (input) are positive.

2. The values of y (output) are also positive.

3. The values of y increase at a decreasing rate as x increases.

Explanation:

A logarithmic function is the inverse of an exponential function, and it represents the relationship between a number and its logarithm. In the given form, b is the base of the logarithm and x is the exponent. In order to understand the properties present in a table representing this function, we need to first understand the concept of logarithms and their properties.

One important property of logarithms is that the input (x) must be a positive number. This is because the logarithm of a negative number is undefined. Therefore, in the given form, the values of x must be positive.

Furthermore, the values of y in a logarithmic function are also positive. This is because the logarithm of a number is always positive, regardless of the base. In the given form, y=logb^x, the values of y will increase as x increases. However, the rate of increase will be decreasing. This means that as x gets larger, the values of y will increase at a slower rate. This can also be seen in the table, where the values of y increase by smaller increments as x increases.

To further understand this concept, let's take an example. Consider a table representing the logarithmic function y=log2^x. As x increases from 1 to 2, the value of y increases from 0 to 1. However, as x increases from 2 to 3, the value of y increases by only 0.5. This trend continues as x increases, with the value of y increasing at a decreasing rate.

In conclusion, a table representing a logarithmic function in the form y=logb^x when b>1 will have the properties of positive values for both x and y, and a decreasing rate of increase for y as x increases. These properties are essential in understanding and graphing logarithmic functions, and they can be seen in the table by examining the values of x and y.

Answer 2

The table is missing, however this can be solved by exploring all properties of a logarithmic function. First of all, we have to consider that properties for logarithmic functions are the same as actual logarithms.

A logarithmic function is the inverse of a exponential function, this means that it's closely related to exponent properties, which are:

`x^(m) x^(n)=x^(m+n) \\(x^(m) )/(x^(n) )=x^(m-n) \\(x^(m))^(n)`

Another important properties are directly related with logarithms:

• Logarithm of a product:
  `log_(a)(MN)=log_(a)M+  log_(a)N`
• Logarithm of a quotient:
  `log_(a)(M)/(N)=log_(a)M-log_(a)N`
• Logarithm of a power:
  `log_(a)M^(n)=n.log_(a)M`

If you compare these 6 properties you will find similarities that help students to memorize them easily. For example, you can say that product related to adding, division relates to subtracting and power relates to multiplying.

Lastly, another important property is actually the relation between logarithmic and exponential functions:

`y=a^(x) \equiv log_(a)y=x`

So, there you have all needed properties to analyse, operate and transform logarithmic functions.