Why is zero not in the doamin of the logarithmic functions y= log3 x and y = log5 x?
because log(0) is undefined, regardless of the base.
There is no power of 3 or 5 which equals zero.
Take a look at the graph of log x.
The reason why zero is not in the domain of the logarithmic functions y = log3 x and y = log5 x is because logarithms are undefined for non-positive numbers.
In these two specific logarithmic functions, the base of the logarithm is either 3 or 5. This means that the logarithm is asking the question "What power do we need to raise the base (3 or 5) to in order to get the input value x?"
For the function y = log3 x, we are asking "What power of 3 do we need to get x?" However, if x is zero or negative, there is no power of 3 that can produce that number. Therefore, zero and all negative numbers are not in the domain of y = log3 x.
Similarly, for the function y = log5 x, we are asking "What power of 5 do we need to get x?" Again, if x is zero or negative, there is no power of 5 that can produce that number. Therefore, zero and all negative numbers are not in the domain of y = log5 x.
In summary, zero is not in the domain of y = log3 x and y = log5 x because logarithms are undefined for non-positive numbers.
To understand why zero is not in the domain of logarithmic functions like y = log3 x and y = log5 x, we need to recall the basic properties of logarithms.
Logarithmic functions are the inverses of exponential functions. In other words, if we have an exponential function in the form y = b^x, then the corresponding logarithmic function is y = logb x.
The domain of a logarithmic function is the set of all possible input values for which the function is defined. In the case of logarithmic functions with a base greater than 1, like log3 x and log5 x, the domain consists of all positive real numbers. This means that zero and negative numbers are not part of the domain.
To see why, let's consider the base 3 logarithmic function, y = log3 x, as an example. The logarithm of a number x to the base 3 is the exponent to which we must raise 3 to get x. In other words, log3 x = y means that 3^y = x.
If we try to find the logarithm of zero in this function, we would have to solve the equation 3^y = 0. However, there is no exponent y that we can raise 3 to in order to get zero. In fact, any positive number raised to any exponent will always result in a positive number. Therefore, zero is not in the domain of the logarithmic function y = log3 x.
The same reasoning applies to the base 5 logarithmic function, y = log5 x. No positive number raised to any exponent will give us zero, so zero is also not in the domain of y = log5 x.
In summary, zero is not in the domain of logarithmic functions with a base greater than 1, like y = log3 x and y = log5 x, because there is no way to obtain zero by raising the base to any exponent.