Problem 86 - Calculus for Biology and Medicine

Question

Write the following expressions in terms of base \(e\) : (a) \(\log _{2}\left(x^{2}-1\right)\) (b) \(\log _{3}(5 x+1)\) (c) \(\log (x+2)\) (d) \(\log _{2}\left(2 x^{2}-1\right)\)

Step-by-Step Solution

Answer

Use the change of base formula to convert each logarithm to base \(e\): (a) \(\frac{\ln(x^2 - 1)}{\ln(2)}\), (b) \(\frac{\ln(5x+1)}{\ln(3)}\), (c) \(\frac{\ln(x+2)}{\ln(10)}\), (d) \(\frac{\ln(2x^2 - 1)}{\ln(2)}\).

Step 1: Understanding Logarithmic Conversion

To convert logarithmic expressions with any base to a natural logarithm (base \(e\)), we use the change of base formula. The formula is as follows: \[\log_a(b) = \frac{\ln(b)}{\ln(a)}.\] We'll use this formula to convert each expression.

Step 2: Convert Expression (a)

For the expression \(\log_{2}(x^{2}-1)\), apply the change of base formula:\[\log_2(x^2 - 1) = \frac{\ln(x^2 - 1)}{\ln(2)}.\] This is the expression in terms of the natural logarithm.

Step 3: Convert Expression (b)

For the expression \(\log_{3}(5x+1)\), apply the change of base formula:\[\log_3(5x+1) = \frac{\ln(5x+1)}{\ln(3)}.\] This converts it to base \(e\).

Step 4: Convert Expression (c)

The expression \(\log(x+2)\) represents a common logarithm (base 10). Convert it using the change of base formula:\[\log(x+2) = \frac{\ln(x+2)}{\ln(10)}.\] Now it’s expressed in terms of the natural logarithm.

Step 5: Convert Expression (d)

Finally, for the expression \(\log_2(2x^2 - 1)\), apply the formula:\[\log_2(2x^2 - 1) = \frac{\ln(2x^2 - 1)}{\ln(2)}.\] This is now in terms of base \(e\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm is a unique and important type of logarithm. It is the logarithm with base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. \ Natural logarithms are denoted as \(\ln(x)\). They are frequently used in many fields like mathematics, physics, and engineering. \ Here's why they're so widely applicable:

• The base \(e\) has natural growth properties that are observed in continuously compounding processes, like population growth or radioactive decay.
• They simplify processes involving exponential growth and decay in calculus since the derivative of \(\ln(x)\) is \(\frac{1}{x}\).

\ Whenever you see a logarithmic expression, converting it to a natural logarithm can help simplify the calculation or allow a deeper understanding of its application in real-life scenarios.

What Are Logarithmic Expressions?

Logarithmic expressions involve logarithms, which are the inverses of exponentials. If you have \(b^y = x\), then the logarithmic form is \(\log_b(x) = y\). \ They express how many times you need to multiply the base \(b\) to get the value \(x\). For example, \(\log_2(8) = 3\), because you need to multiply 2 by itself three times to get 8. \ Logarithmic expressions can seem complex, but they're just a different way to look at multiplication and division:

• They make dealing with very large or very small numbers more manageable.
• They're used to solve equations involving exponentials, known as exponential equations.

\ In our example, we use logarithmic expressions to convert different bases to base \(e\), helping you work with natural logarithms smoothly.

Converting Bases: The Change of Base Formula

Converting between different bases is an essential skill in understanding logarithmic expressions. The change of base formula is a neat tool to convert any logarithm into a different base, usually to base \(e\) for natural logarithms. The formula is: \[\log_a(b) = \frac{\ln(b)}{\ln(a)}\] Here's how it works:

• \(\ln(b)\) computes the natural logarithm of \(b\).
• \(\ln(a)\) computes the natural logarithm of \(a\).
• Dividing these gives you the equivalent logarithmic value but in terms of base \(e\).

This formula is invaluable because:

• It allows you to convert any logarithm to natural logarithms, which are easier to manipulate mathematically.
• Natural logarithms provide a unified language for all sorts of calculations, making them particularly useful in calculus.
• They help you make sense of different scales, especially in scientific phenomena involving exponential relationships.

Mastering the change of base formula will give you a crucial advantage in mathematical analysis, especially when tackling complex logarithmic and exponential problems.