Question

Graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{5} x\)

Short answer

Rewrite \(y = \text{log}_5(x)\) using base 10: \(y = \frac{\text{log}_{10}(x)}{\text{log}_{10}(5)}\). Graph this function using a graphing utility.

Step-by-step solution

Understand the Change-of-Base Formula

The Change-of-Base Formula allows any logarithm with base 'a' to be expressed in terms of logarithms with a different base, usually base 10 or base 'e'. The formula is: \[\text{log}_a(b) = \frac{\text{log}_c(b)}{\text{log}_c(a)}\]where 'c' can be any positive number (commonly 10 or 'e').

Apply the Change-of-Base Formula

Rewrite the given function \(y = \text{log}_5(x)\) using the Change-of-Base Formula with base 10:\[y = \frac{\text{log}_{10}(x)}{\text{log}_{10}(5)}\]. Since \(\text{log}_{10}(5)\) is a constant, this simplifies to:\[y = \frac{\text{log}_{10}(x)}{\text{log}_{10}(5)}\].

Use the Graphing Utility

Now, use a graphing utility to graph the function. Input the transformed function \(y = \frac{\text{log}_{10}(x)}{\text{log}_{10}(5)}\) in the graphing calculator or software. The graph will show the relationship between 'x' and 'y' for the logarithmic function with base 5.

Interpret the Graph

Observe the graph's characteristics. Notice that the curve passes through the point (1,0) because \(\text{log}_5(1) = 0\). The graph also shows the logarithmic growth: as 'x' increases, 'y' increases slowly.

Key concepts

logarithmic functions

Logarithmic functions are mathematical functions that help us understand phenomena such as compound interest, population growth, and more. The general form is \(y = \text{log}_a(x)\), where 'a' is the base and 'x' is the argument. The base can be any positive number except 1. Logarithmic functions are the inverse of exponential functions. This means if \(a^y = x\), then \(y = \text{log}_a(x)\). These functions are very useful for transforming multiplicative relationships into additive ones.
Understanding logarithmic functions often requires familiarity with the Change-of-Base Formula, which simplifies logs to more manageable bases like 10 or 'e'. This is important because most calculators can only compute base 10 (common log) or base 'e' (natural log) by default. Applying the Change-of-Base Formula makes it easier to graph and analyze logs with unconventional bases.

graphing utility

Graphing utilities are tools such as graphing calculators or graphing software that assist in visualizing functions, including logarithmic ones. These tools allow you to input equations and instantly see their graphs. For the function \(y = \text{log}_5(x)\), we can rewrite it using the Change-of-Base Formula to \(y = \frac{\text{log}_{10}(x)}{\text{log}_{10}(5)}\).
By using a graphing utility:

• Input the transformed function into the calculator or software.
• Observe the plot generated by the utility.
• Identify key points like where the function crosses the x-axis.

Graphing utilities provide an intuitive way to see how logarithmic functions behave over different domains and ranges. This dynamic visualization helps in understanding the slow increase synonymous with logarithmic growth.

logarithmic growth

Logarithmic growth describes a scenario where a quantity increases, but the rate of increase slows down over time. This type of growth is common in natural processes and can be seen in many real-world applications like population growth, radioactive decay, and even in the spread of diseases.
For instance, the graph of the logarithmic function \(y = \text{log}_5(x)\) shows that:

• At small values of 'x', 'y' increases quickly.
• As 'x' gets larger, the rate of increase in 'y' becomes slower.

This characteristic slow-growth pattern illustrates why logarithms are useful in fields like demography and electronics, where understanding how variables grow or shrink over time is crucial. It helps us understand that even though 'x' may grow extensively, 'y' grows in a much more controlled and predictable manner.