Problem 5 - Intermediate Algebra

Question

Fill in the blanks. In problem \(6,\) also give the name of each rule. a. \(\log _{b} 1=\square\) b. \(\log _{b} b=\square\) c. \(\log _{b} b^{x}=\square\) \(b^{\log _{b} x}=\square\)

Step-by-Step Solution

Answer

a. 0 (Logarithm of One) b. 1 (Identity Property) c. x (Exponent Rule) \(x\) (Inverse Property)

Step 1: Understanding the Logarithm of One

The problem \( a. \, \log_{b} 1 = \square \) asks us to find the logarithm of 1 for any base \( b \). According to the properties of logarithms, we know that \( \log_b 1 = 0 \) because any number raised to the power of zero equals 1. This is known as the "Logarithm of One" property.

Step 2: Using the Identity Property

The problem \( b. \, \log_{b} b = \square \) is a straightforward application of the identity property of logarithms. For any logarithmic base \( b\), the logarithm of \( b\) itself is always 1: \[ \log_b b = 1. \] This is called the "Identity Property of Logarithms".

Step 3: Applying the Exponent Rule

For \( c. \, \log_{b} b^x = \square \), we use the power rule for logarithms. This rule states that \( \log_b (b^x) = x \) because the logarithm and the exponential function are inverse operations. Therefore, when the base of the logarithm matches the base of the exponent, the exponent \( x \) becomes the result. This is known as the "Exponent Rule" or sometimes referred to as the "Power Rule".

Step 4: Connecting Inverse Functions

For the last equation \( b^{\log_b x} = \square \), it showcases the property where an exponential and a logarithm with the same base cancel each other out, leaving only the argument of the logarithm: \[ b^{\log_b x} = x \] This demonstrates the inverse relationship between exponential functions and logarithms and is often called the "Inverse Property of Exponents and Logarithms."

Key Concepts

Properties of Logarithms

Logarithms have several important properties that simplify complex mathematical expressions. These properties help perform operations involving logs more efficiently and make it easier to solve equations. Here are the key properties you should know:

• Product Property: States that the logarithm of a product is the sum of the logarithms of the factors. That is, \( \log_b(xy) = \log_b x + \log_b y \).


• Quotient Property: States that the logarithm of a quotient is the difference of the logarithms. So, \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \).


• Power Property (Exponent Rule): States that the logarithm of an exponentiated number is the power times the logarithm of the base. Meaning, \( \log_b(x^p) = p \cdot \log_b x \).


• Logarithm of One: For any base \( b \), \( \log_b 1 = 0 \) because any number raised to the power of zero equals 1.


• Identity Property: States that for any base \( b \), \( \log_b b = 1 \).

Identity Property

The identity property of logarithms is one of the most straightforward principles you'll encounter. It asserts that for any logarithmic base \( b \), the logarithm of the base itself is always 1. This can be expressed as:

• \( \log_b b = 1 \)

This is because any number raised to the power of 1 is the number itself. This property serves as a fundamental building block in solving more complex logarithmic expressions.
When you see a logarithm where the base and the number inside the log are the same, you can immediately write down 1. This simplification is often a crucial step in solving logarithmic equations and ensuring they become more manageable.

Exponent Rule

The exponent rule, often referred to as the power rule for logarithms, is a handy tool for dealing with logarithmic expressions of powers. It tells us that:

• \( \log_b(b^x) = x \)

Why does this work? It's because logarithms and exponents are inverse operations. When the base of the logarithm matches the base of the exponent, the power \( x \) is the result.
This property is especially useful in calculus and other advanced mathematics when simplifying expressions or solving equations where powers are involved. It allows one to "bring down" the exponent, making the equation simpler and more straightforward to solve.

Inverse Functions

Logarithms and exponents are known as inverse functions, meaning they undo each other's operations. This inverse relationship is core to understanding how the two functions interact. Consider the equation:

• \( b^{\log_b x} = x \)

Here, the logarithm calculates the power needed for the base \( b \) to become \( x \), and the exponential operation effectively "undoes" this calculation by raising \( b \) to that power, returning the original value \( x \).
This property demonstrates the seamless back-and-forth nature between logarithmic and exponential expressions, which is foundational in algebra and calculus. Recognizing them as inverses helps in solving equations where one cancels out the operation of the other, leading to simplifications and solutions easily.