Question

Transform the following functions to their natural logarithmic forms: (a) \(t=\log _{7} y\) \((b) t=\log _{8}(3 y)\) \((c) t=3 \log _{15}(9 y)\) \((d) t=2 \log _{10} y\)

Short answer

Convert each logarithm using the change of base formula and simplify: (a) \(t = \frac{\ln(y)}{\ln(7)}\), (b) \(t = \frac{\ln(3) + \ln(y)}{\ln(8)}\), (c) \(t = \frac{3 \ln(9) + 3 \ln(y)}{\ln(15)}\), (d) \(t = \frac{2 \ln(y)}{\ln(10)}\)."

Step-by-step solution

Convert log to base 10 form - part A

For the function given in part A, \( t = \log_{7}(y) \), use the change of base formula which states that \( \log_{b}(x) = \frac{\log_{a}(x)}{\log_{a}(b)} \). We convert this to natural log (base \( e \)) using:\[t = \frac{\ln(y)}{\ln(7)}.\]

Convert log to base 10 form - part B

For the function in part B, \( t = \log_{8}(3y) \), apply the change of base formula:\[t = \frac{\ln(3y)}{\ln(8)}\]Use the property of logarithms \( \ln(ab) = \ln(a) + \ln(b) \) to expand the natural log:\[t = \frac{\ln(3) + \ln(y)}{\ln(8)}.\]

Convert log to base 10 form - part C

In part C, the function is \( t = 3 \log_{15}(9y) \). First, apply the change of base formula:\[t = 3 \times \frac{\ln(9y)}{\ln(15)}.\]Split the logarithm using \( \ln(ab) = \ln(a) + \ln(b) \):\[t = 3 \times \frac{\ln(9) + \ln(y)}{\ln(15)}.\]Distribute the factor of 3:\[t = \frac{3 \ln(9) + 3 \ln(y)}{\ln(15)}.\]

Convert log to base 10 form - part D

For part D, where \( t = 2 \log_{10}(y) \), change to natural log form:\[t = 2 \times \frac{\ln(y)}{\ln(10)}.\]Distribute the factor of 2:\[t = \frac{2 \ln(y)}{\ln(10)}.\]

Key concepts

Natural Logarithms

Natural logarithms use the base of Euler's number, denoted as \( e \), which is approximately 2.71828. They are commonly written as \( \ln(x) \) rather than \( \log_{e}(x) \). This type of logarithm is very useful in advanced mathematics, particularly in calculus, because of the special properties of \( e \), such as the derivative of \( e^x \) being \( e^x \), which makes calculations involving growth and decay much simpler.
In our current context, using natural logarithms allows us to transform logarithmic expressions that are typically presented with other bases. By converting to natural logarithms, these expressions can be more easily manipulated with the properties of logarithms.
Overall, natural logarithms are indispensable in higher mathematics, especially in fields that require the modeling of naturally occurring exponential growth processes. They simplify a variety of problems that would otherwise be more complex using other log bases.

Change of Base Formula

The change of base formula is a crucial tool when dealing with logarithms. It allows us to convert a logarithm from one base to another, typically to a more convenient base such as 10 or \( e \). The formula is given by:
\[\log_{b}(x) = \frac{\log_{a}(x)}{\log_{a}(b)}\]
This formula helps in situations where calculators or software may not support a specific base directly. Such adaptability widens our computational resources, owing to the ease with which a log in any base can be expressed through familiar base-10 (common logs) or natural logs.
For converting to natural logarithms, the change of base formula predominantly comes into play as the first step in any expression transformation. It sets the foundation that allows for further simplification using other rules, like the properties of logarithms, which enable us to handle more complex structures within the expression.

Properties of Logarithms

Logarithmic properties are powerful rules that make simplifying and manipulating logarithmic expressions much easier. Some important properties include:

• Product Property: \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \)
• Quotient Property: \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \)
• Power Property: \( \log_{b}(x^c) = c\log_{b}(x) \)

These properties allow for the decomposition of more complex logarithmic expressions into simpler parts. In our exercise, they were used to manage expressions like \( \ln(9y) = \ln(9) + \ln(y) \), breaking down the logarithmic argument into a sum of logarithms.
Understanding these properties also aids in solving logarithmic equations, as they provide multiple ways to approach a problem. This flexibility is invaluable when dealing with the exponential form of relationships and functions, enabling easier transformations and simplifications.

Mathematical Functions

Logarithms are a specific type of mathematical function that is inverse to exponentiation. When we say a function \( f \) is the logarithm of another function, it essentially reverses the operation of that function's base exponent. This characteristic makes logarithms particularly useful for solving equations where the unknown is an exponent.
Mathematical functions, including log functions, are foundational in understanding different patterns and phenomena that occur naturally or in theoretical models. Here, we leverage these functions to model exponential relationships that are common in both natural and social sciences.
Using mathematical functions like logarithms, we transform seemingly complex functions into forms that are more approachable and interpretable. Such transformations allow us to see underlying patterns or achieve desired manipulations that simplify analyses or calculations.