Problem 21
Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{b}\left(x^{2} y\right)$$
Step-by-Step Solution
Verified Answer
The expanded version of the expression \( \log_b(x^{2}y) \) is \( 2*\log_bx + \log_by \)
1Step 1: Apply the product rule
The product rule of logarithm states that the logarithm of the product of two numbers equals the sum of the logarithm of each number separately. So, \( \log_b(x^{2}y) \) can be written as \( \log_bx^{2} + \log_by \).
2Step 2: Apply the power rule
The power rule of logarithm states that the logarithm of a number to an exponent can be written as the product of the exponent and the logarithm of the number. So, \( \log_bx^{2} \) can be written as \( 2*\log_bx \).
3Step 3: Rewrite the expression
The expanded logarithmic expression would then be: \( 2*\log_bx + \log_by \).
Key Concepts
Product Rule of LogarithmsPower Rule of LogarithmsLogarithmic ExpressionsExpanding Logarithms
Product Rule of Logarithms
Logarithms, the inverse of exponentiation, are essential to understanding exponential growth, decay, and many phenomena in science and engineering. Among their most important properties is the product rule, a rule that simplifies the multiplication of numbers. When two or more numbers are multiplied together, their logarithms can be added together if they have the same base.
To use the product rule, one simply rewrites the logarithm of a product, such as \( \text{log}_b(xy) \), as the sum of the logarithms of each individual factor, resulting in \( \text{log}_b(x) + \text{log}_b(y) \). This turns a potentially complex multiplication problem into a simpler addition problem which is easier to manage, especially when dealing with exponential numbers or variable expressions.
To use the product rule, one simply rewrites the logarithm of a product, such as \( \text{log}_b(xy) \), as the sum of the logarithms of each individual factor, resulting in \( \text{log}_b(x) + \text{log}_b(y) \). This turns a potentially complex multiplication problem into a simpler addition problem which is easier to manage, especially when dealing with exponential numbers or variable expressions.
Power Rule of Logarithms
The power rule of logarithms offers a way to deal with exponents within logarithmic expressions. If you have an expression such as \( \text{log}_b(x^n) \), the exponent \( n \) can be pulled out in front of the logarithm, transforming the expression into \( n \times \text{log}_b(x) \).
This property makes it easier to work with logarithmic expressions that involve powers. It's particularly useful because it breaks down the complexity of exponential functions, allowing for straightforward multiplication instead of dealing with harder exponential terms. This rule significantly simplifies the process of solving or manipulating equations involving logarithms.
This property makes it easier to work with logarithmic expressions that involve powers. It's particularly useful because it breaks down the complexity of exponential functions, allowing for straightforward multiplication instead of dealing with harder exponential terms. This rule significantly simplifies the process of solving or manipulating equations involving logarithms.
Logarithmic Expressions
Logarithmic expressions are representations that involve logarithms. These expressions may seem daunting at first, but by understanding the rules of logarithms, they can be simplified or rewritten into more manageable forms. A logarithmic expression, such as \( \text{log}_b(x) \), essentially asks the question: 'To what power do we raise base \( b \) to get \( x \)?'
Understanding how to manipulate these expressions with the product, power, and quotient rules allows students to expand, condense, and solve logarithmic equations. It is crucial to be comfortable converting between exponential and logarithmic form, as this skill frequently comes in handy when evaluating or expanding logarithms without the use of a calculator.
Understanding how to manipulate these expressions with the product, power, and quotient rules allows students to expand, condense, and solve logarithmic equations. It is crucial to be comfortable converting between exponential and logarithmic form, as this skill frequently comes in handy when evaluating or expanding logarithms without the use of a calculator.
Expanding Logarithms
Expanding logarithms is the process of breaking down a complex logarithmic expression into simpler parts by using various logarithmic properties such as the product rule, quotient rule, and power rule. Expansion makes it possible to turn a single log of a large expression into multiple logs, each containing simpler terms.
For instance, expanding \( \text{log}_b(x^2y) \) involves using the product rule to separate the logarithm of a product into the sum of two logarithms, and then applying the power rule to handle the exponent in \( x^2 \). The final expanded form in this case is \( 2 \times \text{log}_b(x) + \text{log}_b(y) \), which is more straightforward and easier to work with for further algebraic manipulation or evaluation.
For instance, expanding \( \text{log}_b(x^2y) \) involves using the product rule to separate the logarithm of a product into the sum of two logarithms, and then applying the power rule to handle the exponent in \( x^2 \). The final expanded form in this case is \( 2 \times \text{log}_b(x) + \text{log}_b(y) \), which is more straightforward and easier to work with for further algebraic manipulation or evaluation.
Other exercises in this chapter
Problem 20
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 8^{1-x}=4^{x+2} $$
View solution Problem 20
In Exercises 9–20, write each equation in its equivalent logarithmic form. $$ 8^{y}=300 $$
View solution Problem 21
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ e^{x+1}=\frac{1}{e} $$
View solution Problem 21
In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{4} 16 $$
View solution