Consider the expression logbase5 a=n
a) For what value(s) of a will n be a positive number?
b) For what value(s) of a will n be a negative number?
c) For what value(s) of a will n be undefined?
To determine the answers to these questions, we need to understand the properties of logarithms and how they relate to positive, negative, and undefined values.
In general, a logarithm function takes the base, which is a positive number, and raises it to a certain power to equal the argument (the number inside the logarithm). Mathematically, it can be written as log_base(x) = y, where x is the argument, y is the exponent or power, and the base is denoted as the subscript of the logarithm.
Now, let's apply these concepts to the given expression, log_base5 a = n, and answer each question:
a) For what value(s) of a will n be a positive number?
For n to be positive, we need the logarithm to be defined. This means that both the base and the argument (a) must be positive numbers. In our case, the base is 5, which is positive. Therefore, for n to be positive, the value of a should be any positive real number greater than zero.
b) For what value(s) of a will n be a negative number?
In this case, n cannot be a negative number because logarithms involving positive bases cannot produce a negative result. Therefore, there are no values of a for which n will be negative.
c) For what value(s) of a will n be undefined?
Logarithms are undefined when the argument or base is zero or negative. In our case, the base is 5, which is positive, so it cannot be zero or negative. Therefore, this expression is defined for all positive real values of a, and there are no undefined values for n.
To summarize:
a) n will be positive for all positive values of a.
b) n will never be negative in this expression.
c) There are no values of a for which n will be undefined.