Problem 273
Question
For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \log _{5} \sqrt{125 x y^{3}} $$
Step-by-Step Solution
Verified Answer
Express: \(\frac{3}{2} + \frac{1}{2} \log _{5} x + \frac{3}{2} \log _{5} y\).
1Step 1: Express the Square Root as an Exponent
The expression inside the logarithm is a square root, which can be expressed as an exponent of 1/2. So, rewrite \( \log _{5} \sqrt{125 x y^{3}} \) as \( \log _{5} (125 x y^{3})^{1/2} \).
2Step 2: Apply the Power Rule
Using the power rule of logarithms, \( \log_b(A^n) = n\log_b(A) \), we can simplify: \[ \log _{5} (125 x y^{3})^{1/2} = \frac{1}{2} \log _{5} (125 x y^{3}). \]
3Step 3: Apply the Product Rule
The expression \( 125 x y^{3} \) is a product. Using the product rule for logarithms, \( \log_b(ABC) = \log_b A + \log_b B + \log_b C \), we have: \[ \frac{1}{2} (\log _{5} 125 + \log _{5} x + \log _{5} y^{3}). \]
4Step 4: Apply the Power Rule Again
Use the power rule on \( \log _{5} y^{3} \) to get \( 3 \log _{5} y \). Therefore: \[ \frac{1}{2} (\log _{5} 125 + \log _{5} x + 3 \log _{5} y). \]
5Step 5: Simplify Logarithms where Possible
If further simplification of \( \log _{5} 125 \) is possible, express it using the same base: \( 125 = 5^3 \), so \( \log _{5} 125 = 3 \). Thus, the expression becomes: \[ \frac{1}{2} (3 + \log _{5} x + 3 \log _{5} y). \]
Key Concepts
Logarithm Product RuleLogarithm Power RuleLogarithm Simplification
Logarithm Product Rule
The logarithm product rule is a fantastic tool to break down more complex expressions. It states that the logarithm of a product is equal to the sum of the logarithms of its factors:
\[ \log_b(ABC) = \log_b A + \log_b B + \log_b C \]
For example, if you have \( \log_b(125xy^3) \), you can apply the product rule to rewrite it as \( \log_b 125 + \log_b x + \log_b y^3 \).
This is useful because it turns a single complicated expression into a set of simpler expressions. By breaking it down, you can further apply other rules like the power rule, making each part more manageable.
In essence, the product rule helps you simplify expressions by dividing the work into smaller, easier-to-handle pieces.
\[ \log_b(ABC) = \log_b A + \log_b B + \log_b C \]
For example, if you have \( \log_b(125xy^3) \), you can apply the product rule to rewrite it as \( \log_b 125 + \log_b x + \log_b y^3 \).
This is useful because it turns a single complicated expression into a set of simpler expressions. By breaking it down, you can further apply other rules like the power rule, making each part more manageable.
In essence, the product rule helps you simplify expressions by dividing the work into smaller, easier-to-handle pieces.
Logarithm Power Rule
The logarithm power rule comes in handy when dealing with exponents inside a logarithm. It helps simplify expressions by allowing you to "bring down" the exponent in front of the logarithm:
\[ \log_b(A^n) = n \log_b A \]
Let's see it in action with \( \log_5(y^3) \); using this rule, it becomes \( 3 \log_5 y \).
This makes the expression easier to compute, particularly when evaluating logarithms of terms raised to powers.
With the power rule, you can tackle even large or complicated exponents methodically, simplifying your task.
This process helps focus on one component at a time, making sure each part of the expression is as easy to interpret as possible.
\[ \log_b(A^n) = n \log_b A \]
Let's see it in action with \( \log_5(y^3) \); using this rule, it becomes \( 3 \log_5 y \).
This makes the expression easier to compute, particularly when evaluating logarithms of terms raised to powers.
- Pull the exponent in front, making it a coefficient.
- Use this after applying the product rule to manage each log term independently.
With the power rule, you can tackle even large or complicated exponents methodically, simplifying your task.
This process helps focus on one component at a time, making sure each part of the expression is as easy to interpret as possible.
Logarithm Simplification
Simplification is the final polish in handling logarithms, often using knowledge of specific logarithmic values or identities.
For example, one common simplification is \( \log_5(125) \). Since 125 is \( 5^3 \), replacing it leads to the calculation \( \log_5 125 = 3 \).
Breaking the expression into reachable figures aids in utilizing known values efficiently. It reduces complex expressions to simple numbers.
Whether reducing a large number by recognizing its base relation or computing simple logs through properties, simplification is the finishing step.
It ties together previous rules like the product and power rules to provide an understandable and elegant result.
For example, one common simplification is \( \log_5(125) \). Since 125 is \( 5^3 \), replacing it leads to the calculation \( \log_5 125 = 3 \).
Breaking the expression into reachable figures aids in utilizing known values efficiently. It reduces complex expressions to simple numbers.
- Calculate exact values when possible.
- Use known logarithmic identities to simplify.
Whether reducing a large number by recognizing its base relation or computing simple logs through properties, simplification is the finishing step.
It ties together previous rules like the product and power rules to provide an understandable and elegant result.
Other exercises in this chapter
Problem 272
For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \ln a^{3} \sqrt{b} $$
View solution Problem 272
Use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. \(\ln a \sqrt[3]{b}\)
View solution Problem 273
Use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. \(\log _{5} \sqrt{125 x y^{3}}\)
View solution Problem 274
For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \log _{4} \frac{\sqrt[
View solution