Problem 141 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } \log (x+3)=2, \text { then } e^{2}=x+3 $$

Step-by-Step Solution

Answer

The statement is false. The corrected statement is 'If \( \ln(x+3)=2 \), then \( e^{2}=x+3 \)'.

Step 1: Confirm the Base

The base of the logarithm is not explicitly stated. By default, when the base is not specified, it is base 10. In the exponential function, however, the base is \( e \).

Step 2: Change the base of the logarithm

The relation between logarithmic and exponential functions states that for \( b > 0 \) and \( b \neq 1 \), \( b^{\log_b(x)} = x \). As the base of the log in the question is 10 and the base of the exponential function is \( e \), so the statement is false. To make a true statement, the base of the logarithm needs to match the base of the logarithm in the exponential form. The true statement is \( \ln(x+3)=2 \), then \( e^{2}=x+3 \). In the corrected form, a natural log (\( \ln \)) which has base \( e \), has been used to match the base in the exponential form.

Key Concepts

Understanding Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions often appear in the form of \( b^x \), where \( b \) is the base and \( x \) is the exponent. They are widely used to model growth and decay processes, such as population growth or radioactive decay.

One of the key characteristics of exponential functions is their rapid change. As the exponent increases, the value of the function rises sharply. This can be seen when plotting an exponential curve, which shows a distinct upward or downward "J" shape depending on whether the base \( b \) is greater than 1 or between 0 and 1.

In the context of solving equations, exponential functions can be rewritten in logarithmic form to make them easier to manipulate. This involves understanding the relationship between the two types of mathematical expressions, which we will explore further in the next sections.

How to Approach Base Conversion

Base conversion in mathematics often involves transforming an expression or function from one base to another to simplify calculations or make the problem easier to solve. In the realm of logarithms and exponential functions, it's crucial to ensure that both sides of an equation have the same base to be accurate. This is particularly important when one side of the equation contains a logarithm, and the other side contains an exponential expression.

When given a problem where the bases don't match, you'll need to convert the base of one expression. A common method is to convert a given log with an arbitrary base to one with a natural log using the change of base formula:

• \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \)

This formula allows you to switch between different log bases easily and ensures consistency, especially when working with the natural number \( e \), an important constant in mathematics.
In the scenario from the exercise, the statement contained a misunderstanding due to mismatched bases, which was corrected by using natural logarithms to match the base \( e \) in the exponential function.

Exploring Natural Logarithms

Natural logarithms, denoted as \( \ln(x) \), use Euler's number \( e \) as their base. The constant \( e \) is approximately equal to 2.718 and is an irrational number that arises frequently in continuous growth and decay processes.

Natural logs are particularly useful in mathematical calculus, as they simplify differentiation and integration. When you differentiate \( \ln(x) \), the derivative is simply \( \frac{1}{x} \). This property makes \( \ln(x) \) a preferred choice in many advanced mathematical contexts.

In practical applications, natural logs often allow us to model complex systems naturally and intuitively, such as compound interest in finance or varying rates of population change in biology. Whenever you see expressions involving \( e \), it might be beneficial to use natural logs to simplify the process.
Referring back to the exercise, using the correct natural logarithm ensured the bases were aligned, allowing for an accurate solution to the equation.