Problem 32 - Precalculus Mathematics for Calculus

Question

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{4} 2=x\) (b) \(\log _{4} x=2\)

Step-by-Step Solution

Answer

(a) \(x = \frac{1}{2}\) (b) \(x = 16\)

Step 1: Understanding Logarithmic Form

The logarithmic equation \ \(\log_b a = x\ \) implies that \ \(b^x = a\ \). This is the definition of a logarithm, stating that the logarithm of \(a\) with base \(b\) is \(x\), when \(b\) raised to the power of \(x\) equals \(a\).

Step 2: Solving Part (a)

For equation (a), \(\log_4 2 = x\), we apply the logarithmic definition: \(4^x = 2\). To find \(x\), observe that \(2 = 4^{1/2}\), or equivalently \(2 = (2^2)^{1/2} = 2^{1}\). This simplifies to \(4^{1/2} = 2\), indicating \(x = \frac{1}{2}\).

Step 3: Solving Part (b)

For equation (b), \(\log_4 x = 2\), we again utilize the logarithmic definition: \(4^2 = x\). Calculating \(4^2\) yields \(16\). Thus, \(x = 16\).

Key Concepts

Exponential Form

Exponential form is a mathematical way of expressing numbers through powers and bases. When you come across something like \(b^x = a\), it means you're dealing with the exponential form. Here, \(b\) is the base and \(x\) is the exponent. The core idea is that the base raised to the power of the exponent equals \(a\). This concept is fundamental not only in understanding logarithms but also in a wide range of mathematical applications, such as solving equations and modeling growth scenarios.

In the context of logarithms, understanding exponential form is crucial. When you have a logarithmic equation like \(\log_b a = x\), it's saying that the base \(b\), when raised to the power \(x\), gives you \(a\).

• This relationship helps to "unwrap" the log back into its exponential expression.
• This conversion is essential for solving logarithmic equations, as it often makes them much more straightforward to solve.

Having a strong grasp of exponential form allows you to move smoothly between interpreting logarithmic statements and solving practical problems.

Logarithm Equations

Logarithm equations are a way to find unknown exponents. They occur when we set two logarithmic expressions equal to one another. The essence of a logarithmic equation is that it lets us work backward from an exponential relationship. For instance, in the problem \(\log_4 2 = x\), our task is to figure out what power the base \(4\) must be raised to, in order to yield \(2\).

To handle logarithm equations:

• Convert the logarithmic equation into its corresponding exponential form. For example, converting \(\log_4 2 = x\) gives us \(4^x = 2\).
• Solve the resulting exponential equation. In this case, recognizing that \(2 = 4^{1/2}\) leads directly to \(x = \frac{1}{2}\).

This method ensures a step-by-step process that simplifies tackling problems involving logarithms. By translating a logarithmic equation into its exponential counterpart, it becomes easier to solve for the unknown variable.

Base of Logarithm

The base of a logarithm is the number that is raised to a power to get another number. It's a crucial part of understanding how logarithms work. In a logarithmic expression like \(\log_b a\), \(b\) is the base. Knowing the base is important, because different bases change how you approach solving the equation.

When dealing with different types of bases:

• If \(b = 10\), it's called a common logarithm, often written simply as \(\log\).
• If \(b = e\) (where \(e\) is the Euler's number, approximately 2.71828), it's a natural logarithm, noted as \(\ln\).
• Specific problems can use bases like 2, which often appear in computer science due to binary systems.

The chosen base in a logarithmic equation defines the exponential relationship you will need to work with. For instance, in \(\log_4 x = 2\), \(4\) is the base, which means \(x = 4^2 = 16\). Understanding the base ensures you correctly apply the rules of logarithms in various contexts.