Problem 3 - Intermediate Algebra

Question

Fill in the blanks. a. If two exponential expressions with the same base are equal, their exponents are ___. \(b^{x}=b^{y} \quad\) is equivalent to __=__. b. If the logarithms base-b of two numbers are equal, the numbers are ___. $$ \log _{b} x=\log _{b} y $$ is equivalent to __=__.

Step-by-Step Solution

Answer

a. Exponents are equal: \( x = y \). b. Numbers are equal: \( x = y \).

Step 1: Analyze the Exponential Equation

The equation given is \( b^x = b^y \). This is an equation involving exponential expressions with the same base \( b \). When two exponential expressions with identical bases are equal, it implies that their exponents are equal. This can be written as \( x = y \).

Step 2: Fill in the First Blank for Exponents

For the equation \( b^x = b^y \), since the exponential expressions have the same base, the exponents must be equal. Thus, we fill the blank with 'equal'.

Step 3: Fill in the Second Blank for Exponential Equality

For \( b^x = b^y \), as concluded in Step 1, \( x = y \). So, we fill in the blanks for "is equivalent to __=__" with \( x = y \).

Step 4: Analyze the Logarithmic Equation

The equation given is \( \log_b x = \log_b y \). This involves logarithmic expressions with the same base \( b \). If the logarithms of two numbers to the same base are equal, the numbers themselves must be equal. This is written as \( x = y \).

Step 5: Fill in the Blank for Logarithmic Equality

For the equation \( \log_b x = \log_b y \), since the logarithms with the same base are equal, the numbers must be equal. Thus, we fill in the blanks for "is equivalent to __=__" with \( x = y \).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are written as a small number, called the exponent, to the upper right of the base number.
This means if you have an expression like \( a^n \), it represents multiplying the number \( a \) by itself \( n \) times.
Here are a few fundamentals about exponents you should know:

• Any number raised to the power of 1 equals the number itself. For example, \( 5^1 = 5 \).
• Any number raised to the power of 0 is 1. So, \( 8^0 = 1 \).
• When multiplying powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing powers with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

Understanding these basic rules helps simplify many exponential equations.

Logarithms

Logarithms are the inverses of exponentials, making them a powerful tool for solving exponential equations.
If you encounter an equation in the form \( b^x = y \), the logarithmic form is expressed as \( x = \log_b y \).
This means that the logarithm asks the question: "To what power do we need to raise \( b \) to get \( y \)?"
Important aspects of logarithms include:

• The logarithm of 1 to any base is always 0: \( \log_b 1 = 0 \) because \( b^0 = 1 \).
• The logarithm of a number equal to its base is always 1: \( \log_b b = 1 \) because \( b^1 = b \).
• Logarithm properties: \( \log_b (mn) = \log_b m + \log_b n \) and \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \).

These properties are key to simplifying and solving equations involving logarithms.

Equations

An equation is a mathematical statement that asserts the equality of two expressions.
Solving an equation involves finding the value of the variable that makes the equation true.
There are different types of equations such as linear, quadratic, exponential, and logarithmic. Each type has specific methods for finding solutions.
Let's look closer at equations like \( b^x = b^y \) and \( \log_b x = \log_b y \):

• For exponential equations with equal bases, the exponents themselves are equal, leading to simpler solutions, such as \( x = y \).
• Similarly, if logarithms with the same base are equal, the arguments of these logarithms must be the same, hence \( x = y \).

Understanding these principles is essential for mastering the solution of various types of equations in mathematics.