Problem 35
Question
Use the Laws of Logarithms to expand the expression. $$\log _{2}\left(\frac{x\left(x^{2}+1\right)}{\sqrt{x^{2}-1}}\right)$$
Step-by-Step Solution
Verified Answer
\( \log_{2}(x) + \log_{2}(x^2+1) - \frac{1}{2} \log_{2}(x^2-1) \)
1Step 1: Apply the Quotient Rule
The expression given is \( \log_{2} \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) \). Use the quotient rule for logarithms: \( \log_{b} \left( \frac{u}{v} \right) = \log_{b}(u) - \log_{b}(v) \). Split the expression into two parts: \( \log_{2}(x(x^2+1)) - \log_{2}(\sqrt{x^2-1}) \).
2Step 2: Apply the Product Rule
Now apply the product rule to the first term: \( \log_{2}(x(x^2+1)) = \log_{2}(x) + \log_{2}(x^2+1) \). The product rule states: \( \log_{b}(uv) = \log_{b}(u) + \log_{b}(v) \).
3Step 3: Simplify using the Power Rule
Consider the second term from Step 1: \( \log_{2}(\sqrt{x^2-1}) \). The square root can be expressed as a power: \( (x^2-1)^{1/2} \). Applying the power rule, \( \log_{b}(u^{c}) = c \cdot \log_{b}(u) \), we get \( \frac{1}{2} \cdot \log_{2}(x^2-1) \).
4Step 4: Combine All Terms
Combine all the expanded parts: \( \log_{2}(x) + \log_{2}(x^2+1) - \frac{1}{2} \log_{2}(x^2-1) \).
Key Concepts
Laws of LogarithmsQuotient RuleProduct RulePower Rule
Laws of Logarithms
Understanding the laws of logarithms is crucial for manipulating and simplifying logarithmic expressions. These laws serve as foundational rules that govern the properties of logarithms, making it easier to expand, condense, or solve equations. Here are the key laws that you need to know:
- Product Rule: This rule helps in splitting the logarithm of a product into a sum of separate logarithms, i.e., \( \log_{b}(uv) = \log_{b}(u) + \log_{b}(v) \).
- Quotient Rule: This rule is used to separate a logarithm of a division into a subtraction of logarithms, i.e., \( \log_{b} \left( \frac{u}{v} \right) = \log_{b}(u) - \log_{b}(v) \).
- Power Rule: This allows for bringing down an exponent in the argument of a logarithm, which simplifies computations: \( \log_{b}(u^{c}) = c \cdot \log_{b}(u) \).
Quotient Rule
The quotient rule is a fundamental tool in logarithmic expressions that allows you to handle the division of terms inside a logarithm. It states that the logarithm of a quotient is equal to the difference of the logarithms: \( \log_{b} \left( \frac{u}{v} \right) = \log_{b}(u) - \log_{b}(v) \). This rule is particularly useful in breaking down complex expressions into more manageable parts.
For example, in the exercise, you start with \( \log_{2} \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) \). Applying the quotient rule, we split this into two separate logarithmic expressions: \( \log_{2}(x(x^2+1)) - \log_{2}(\sqrt{x^2-1}) \). This separation lays the groundwork for further simplification using other rules.
For example, in the exercise, you start with \( \log_{2} \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) \). Applying the quotient rule, we split this into two separate logarithmic expressions: \( \log_{2}(x(x^2+1)) - \log_{2}(\sqrt{x^2-1}) \). This separation lays the groundwork for further simplification using other rules.
Product Rule
The product rule of logarithms simplifies the logarithm of a product into a sum of logarithms. It states: \( \log_{b}(uv) = \log_{b}(u) + \log_{b}(v) \). This rule is extremely helpful when you need to expand logarithmic expressions that involve products.
In the exercise, after applying the quotient rule, we encounter \( \log_{2}(x(x^2+1)) \). Here, we can further break this down using the product rule: \( \log_{2}(x) + \log_{2}(x^2+1) \). This breakdown converts a multiplication inside a logarithm expression into two separate and more straightforward logarithms, thus making our expressions easier to manipulate and solve further down the process.
In the exercise, after applying the quotient rule, we encounter \( \log_{2}(x(x^2+1)) \). Here, we can further break this down using the product rule: \( \log_{2}(x) + \log_{2}(x^2+1) \). This breakdown converts a multiplication inside a logarithm expression into two separate and more straightforward logarithms, thus making our expressions easier to manipulate and solve further down the process.
Power Rule
In logarithms, the power rule is utilized to simplify expressions by managing exponents. The rule is expressed as: \( \log_{b}(u^{c}) = c \cdot \log_{b}(u) \). This essentially allows you to bring the exponent outside the logarithm, simplifying the calculations involved.
Referencing the example from the exercise, after applying the quotient and product rules, we address the term \( \log_{2}(\sqrt{x^2-1}) \). The square root can be rewritten as \((x^2-1)^{1/2} \). Applying the power rule, we transform this into \( \frac{1}{2} \cdot \log_{2}(x^2-1) \). By doing so, you make the expression easier to handle and equip yourself for algebraic manipulations that might follow in more complex exercises.
Referencing the example from the exercise, after applying the quotient and product rules, we address the term \( \log_{2}(\sqrt{x^2-1}) \). The square root can be rewritten as \((x^2-1)^{1/2} \). Applying the power rule, we transform this into \( \frac{1}{2} \cdot \log_{2}(x^2-1) \). By doing so, you make the expression easier to handle and equip yourself for algebraic manipulations that might follow in more complex exercises.
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