Question

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ f(a x)=f(a)+f(x), \quad a>0, x>0 $$

Short answer

The statement 'If \(f(x)=\ln x\), then \(f(a x)=f(a)+f(x)\' where \(a>0, x>0\) is true because it adheres to the logarithmic product rule.

Step-by-step solution

Evaluate the Expression Using the Product Rule

Using the logarithm product rule, substitute \(f(ax)\) with \(\ln(ax)\). Also replace \(f(a)\) and \(f(x)\) with \(\ln(a)\) and \(\ln(x)\) respectively. This results in the following: \(\ln(ax)=\ln(a)+\ln(x)\). This equation is equivalent to the logarithm product rule which states that the log of a product is the sum of the logs of the factors. Therefore the statement \(f(a x)=f(a)+f(x)\), where \(a>0, x>0\) is true.

Check the valid range of x and a

Since the domain of the function \(\ln(x)\) includes all positive real numbers, the values of \(a>0\) and \(x>0\) specified in the problem fall within the domain of the function. Therefore the function is well-defined for these values.

Final verification

It has been established that the equation \(\ln(ax)=\ln(a)+\ln(x)\) holds true for all positive values of \(a\) and \(x\). Therefore, the given expression \(f(a x)=f(a)+f(x)\) for the function \(f(x)=\ln x\) is a valid statement.

Key concepts

Logarithm Product Rule

When delving into the behavior of logarithms, the product rule is an essential property to understand. It states that the logarithm of a product is equal to the sum of the logarithms of its individual factors. Mathematically:

• Given two positive numbers, say, \(a\) and \(x\), the product rule tells us that \(\ln(ax) = \ln(a) + \ln(x)\).

This property is crucial because it simplifies complex multiplication processes into manageable addition tasks. It's particularly handy in algebra for simplifying equations and solving logarithmic expressions efficiently.
This rule applies to all positive numbers since logarithms are only defined for positive real numbers. Understanding the product rule opens up pathways to further explore logarithmic functions and their applications in various mathematical problems.

Function Domain

The domain of a function is a critical aspect that defines the set of inputs for which the function is mathematically valid. For the natural logarithm function \(f(x) = \ln x\), the domain consists exclusively of positive real numbers.

• This means \(x>0\) for any expression involving \(\ln x\) to be valid.
• In the context of the problem, both \(a\) and \(x\) need to be greater than zero to ensure the function can be applied.

Understanding the domain is vital because it prevents mathematical errors that could occur from attempting to compute the logarithm of zero or negative numbers. Ensuring that the inputs adhere to the domain conditions keeps mathematical expressions well-defined and properly valid, justifying the correctness of the product rule in these scenarios.

Natural Logarithm

A natural logarithm is a specific logarithm with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.718. The notation \(\ln x\) specifically refers to the logarithm of \(x\) to the base \(e\).

• Natural logarithms are extensively used in continuous growth and decay problems, as they seamlessly tie into the exponential functions and processes.

In practical terms, when you see \(\ln x\), you're inherently referring to its relationship with exponential functions, such as how it "undoes" exponential processes like calculating continuous growth rates.
In the exercise, the natural logarithm \(\ln\) plays a key role in ensuring the validity of the statement through the product rule, as it requires products to be broken down into sums - a critical aspect of solving exponential and logarithmic equations.