Problem 75 - Calculus for Biology and Medicine

Question

Which real number \(x\) satisfies (a) \(\log _{4} x=-2\) ? (b) \(\log _{1 / 3} x=-3 ?\) (c) \(\log _{10} x=-2\) ?

Step-by-Step Solution

Answer

(a) \(x = \frac{1}{16}\); (b) \(x = 27\); (c) \(x = \frac{1}{100}\).

Step 1: Understand Logarithm Properties

Recall the fundamental property of logarithms: If \( \log_b a = c \), then \( b^c = a \). This property will be used to find the values of \( x \) in each of the problems given.

Step 2: Solve Part (a)

We have \( \log _{4} x=-2 \). Applying the property \( b^c = a \), we have \( 4^{-2} = x \). Thus, \( x = 4^{-2} \), which simplifies to \( x = \frac{1}{16} \).

Step 3: Solve Part (b)

We have \( \log _{1/3} x = -3 \). Using the property \( b^c = a \), we get \( \left( \frac{1}{3} \right)^{-3} = x \). This simplifies to \( x = 3^3 = 27 \).

Step 4: Solve Part (c)

We have \( \log _{10} x = -2 \). According to the property \( b^c = a \), it follows that \( 10^{-2} = x \). This simplifies to \( x = \frac{1}{100} \).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them incredibly useful in mathematics, especially when solving equations. One crucial property is the relationship between a logarithm and exponents: if you have a log in the form of \( \log_b a = c \), it tells you that \( b^c = a \). This property provides a powerful tool for converting between logarithmic and exponential forms, enabling us to solve for unknown values. Other useful properties include:

• Logarithm of a product: \( \log_b (mn) = \log_b m + \log_b n \).
• Logarithm of a quotient: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \).
• Logarithm of a power: \( \log_b (m^n) = n \log_b m \).

These properties allow for the simplification and manipulation of complex logarithmic expressions, making them easier to work with. Understanding these properties is key to mastering logarithms.

Logarithmic Equations

Logarithmic equations involve logarithms and are equations where the unknown variable is inside a logarithm. To solve these types of equations, it's often helpful to use the property that relates logarithms to exponents. For instance, in the equation \( \log_b x = y \), you can rewrite it as \( b^y = x \) to solve for \( x \). This transformation is essential because it converts the equation into a form that many students find easier to work with: an exponential equation.When solving logarithmic equations:

• Always check if you can use the basic property \( b^c = a \) to rewrite the equation.
• Be vigilant about the domain of the logarithm, ensuring that the arguments are positive since logarithms of non-positive numbers are undefined.
• If using properties results in multiple solutions, verify each one to ensure they fit within the allowed domain.

Logarithmic equations may initially seem daunting, but breaking them down using fundamental properties simplifies the process and provides insight into their solutions.

Base of a Logarithm

The base of a logarithm is a critical component that determines the form and properties of the logarithm itself. In a logarithmic expression like \( \log_b a \), \( b \) is the base. Common bases include 10 (common logarithms), \( e \) (natural logarithms), and 2 (binary logarithms), but any positive number except 1 can be a base.Understanding the impact of the base is essential:

• Logarithms with base 10 are used frequently in science and engineering because many products come with powers of 10.
• Natural logarithms, which use base \( e \), are crucial in calculus and natural growth processes because \( e \) is the rate of continuous growth.
• Choosing the appropriate base can simplify calculations, especially if you're dealing with exponential growth or decay that naturally aligns with powers of a certain base.

When working with logarithms, always pay attention to the base, as different bases change the results and may require different approaches to solve equations efficiently.