How to Compare the Graphs of y=log(2x)+3 and y=log(x)

Logarithmic functions are useful for modeling phenomena that involve exponential growth or decay, such as population, radioactive decay, sound intensity, pH, and more. In this article, we will explore how to compare the graphs of two logarithmic functions: y=log(2x)+3 and y=log(x). We will also learn how to use transformations to relate the graphs of these functions.

What is a Logarithmic Function?

A logarithmic function is an inverse function of an exponential function. That means that if we have an exponential function of the form y=a^x, where a is a positive constant, then its inverse function is a logarithmic function of the form x=a^y, or equivalently, y=log_a x. The base a of the logarithm determines how fast the function grows or decays. The most common bases are 10 and e (the natural logarithm).

A logarithmic function has a domain of all positive real numbers and a range of all real numbers. It is always increasing, but at a decreasing rate. It has a vertical asymptote at x=0 and an x-intercept at x=1. The graph of a logarithmic function is shown below.

How to Transform a Logarithmic Function?

We can transform a logarithmic function by applying shifts, stretches, compressions, and reflections to its graph. These transformations affect the equation of the function in different ways. Here are some examples:

• To shift the graph horizontally by h units to the right, we replace x with x-h in the equation. For example, y=log(x-2) is shifted 2 units to the right compared to y=log(x).
• To shift the graph vertically by k units up, we add k to the equation. For example, y=log(x)+3 is shifted 3 units up compared to y=log(x).
• To stretch or compress the graph horizontally by a factor of b, we divide x by b in the equation. For example, y=log(2x) is compressed by a factor of 2 compared to y=log(x).
• To stretch or compress the graph vertically by a factor of c, we multiply the equation by c. For example, y=2log(x) is stretched by a factor of 2 compared to y=log(x).
• To reflect the graph across the x-axis, we multiply the equation by -1. For example, y=-log(x) is reflected across the x-axis compared to y=log(x).
• To reflect the graph across the y-axis, we replace x with -x in the equation. For example, y=log(-x) is reflected across the y-axis compared to y=log(x).

How to Compare the Graphs of y=log(2x)+3 and y=log(x)?

Now that we know how to transform a logarithmic function, we can compare the graphs of y=log(2x)+3 and y=log(x). We can use the following steps:

• Identify the base and the transformations applied to each function.
• Sketch the graph of the original function y=log(x) using its key features.
• Apply the transformations to the graph of y=log(x) to obtain the graph of each function.
• Compare and contrast the graphs of each function.

Step 1: Identify the base and the transformations applied to each function.

The base of both functions is 10, since it is not specified in the equation. The transformations applied to each function are:

• For y=log(2x)+3, we have a horizontal compression by a factor of 2 and a vertical shift up by 3 units.
• For y=log(x), we have no transformations.

Step 2: Sketch the graph of the original function y=log(x) using its key features.

The graph of y=log(x) has a domain of (0, infinity), a range of (-infinity, infinity), a vertical asymptote at x=0, an x-intercept at x=1, and an increasing but concave down shape. We can sketch it as follows:

Step 3: Apply the transformations to the graph of y=log(x) to obtain the graph of each function.

To obtain the graph of y=log(2x)+3, we need to compress the graph horizontally by a factor of 2 and shift it up by 3 units. This means that:

• The domain becomes (0, infinity).
• The range becomes (3, infinity).
• The vertical asymptote becomes x=0.
• The x-intercept becomes x=0.5.
• The shape remains increasing but concave down.

We can sketch it as follows:

![Graph of y=log(2x)+3] (https://en.wikipedia.org/wiki/Logarithm)

To obtain the graph of y=log(x), we do not need to apply any transformations. We can use the same graph as in step 2.

Step 4: Compare and contrast the graphs of each function.

The graphs of y=log(2x)+3 and y=log(x) have some similarities and some differences. Some of them are:

• Both graphs have the same base, the same vertical asymptote, and the same shape.
• The graph of y=log(2x)+3 is shifted up by 3 units compared to the graph of y=log(x). This means that it has a higher y-intercept and a higher minimum value.
• The graph of y=log(2x)+3 is compressed horizontally by a factor of 2 compared to the graph of y=log(x). This means that it has a smaller x-intercept and a steeper slope.