Problem 79 - Algebra 2

Question

Find the domain and the range of each function. $$ y=1+\log x $$

Step-by-Step Solution

Answer

The domain of the function is \(x > 0\) and the range is all real numbers (\(-\infty < y < \infty\)).

Step 1: Identify the domain

Content for Step 1: First, we need to clarify that the function \(y=1+\log x \) is only defined for \(x>0\) according to the definition of a logarithm. Therefore, our domain is the set of all positive real numbers, written as \(x > 0\).

Step 2: Find the range

Content for Step 2: \(y = 1 + \log x \) can output any real number, since it simply adds 1 to the logarithm of x. The logarithm of x can be any real number, and adding one doesn't change this, so our range is all real numbers, or \(-\infty < y < \infty\).

Key Concepts

Logarithmic Functions

Logarithmic functions are a type of function that are the inverses of exponential functions. In simpler terms, if you have an exponential function like \( y = a^x \), the logarithmic function would be \( x = \log_a(y) \). Logarithms help us solve for exponents in equations where the exponent is the unknown variable. In the equation \( y=1+\log(x) \), the logarithmic part \( \log(x) \) represents the power to which a base number must be raised to obtain the number \( x \). Logarithms are particularly useful in fields such as

• Science, for dealing with large numerical ranges.
• Engineering, to model complex systems.
• Economics, to track growth rates and predict trends.

In our example, the expression \( \log(x) \) is using the common logarithm, which has a base of 10, unless otherwise stated. Remember, logarithmic functions provide us with a way to map large ranges of values into smaller ones, making them much more manageable in various calculations.

Domain of a Function

The domain of a function refers to all possible input values (usually \( x \) values) that our function can accept. For the logarithmic function \( y=1+\log x \), we need to determine which \( x \) values will make the function a valid operation. For logarithms, you can't take the log of zero or a negative number, because it is undefined in the realm of real numbers. This means that \( \log(x) \) is only defined for \( x>0 \). Therefore, the domain of \( y=1+\log x \) is:

• All positive real numbers \(x>0\).

You write this in interval notation as \((0, \infty)\). It's crucial to always check the domain of a function when working with logarithms, to ensure that calculations remain valid and meaningful.

Range of a Function

The range of a function is the set of all possible output values (usually \( y \) values) that can be produced by a function. In the case of \( y=1+\log x \), we are trying to determine what kind of values \( y \) can be. For logarithmic functions \( \log(x) \), since the logarithm of a number theoretically goes from negative infinity (as \( x \) approaches zero from the positive side) to positive infinity (as \( x \) gets very large), the range is going to cover all real numbers. Adding a constant (like the "1" in our function) to \( \log x \) does not change the range. It simply horizontal shifts the graph of the log function up or down. So \( y=1+\log x \), like \( \log x \), has the range:

• All real numbers \(-\infty < y < \infty\).

The range of logarithmic functions is very useful because it allows us to know that mapping to the function can cover all real number outputs, given permissible inputs.