Problem 80
Question
Write logarithmic expression as one logarithm. \(3 \log _{b}(x+1)-2 \log _{b}(x+2)+\log _{b} x\)
Step-by-Step Solution
Verified Answer
\(\log_{b} \left( \frac{x(x+1)^3}{(x+2)^2} \right)\)
1Step 1: Use the Power Rule
The power rule of logarithms states that if you have a constant multiplying a logarithm, you can rewrite it as an exponent on the term inside the logarithm. Apply this rule:\[ 3 \log_{b}(x+1) = \log_{b}((x+1)^3) \]\[ 2 \log_{b}(x+2) = \log_{b}((x+2)^2) \]
2Step 2: Apply the Quotient Rule
The quotient rule of logarithms states that the subtraction of two logarithms is equivalent to the logarithm of the division of their arguments. Thus:\[ 3 \log_{b}(x+1) - 2 \log_{b}(x+2) = \log_{b} \left( \frac{(x+1)^3}{(x+2)^2} \right) \]
3Step 3: Use the Product Rule
The product rule states that the addition of two logarithms is equivalent to the logarithm of the product of their arguments. Apply this rule:\[ \log_{b} \left( \frac{(x+1)^3}{(x+2)^2} \right) + \log_{b} x = \log_{b} \left( x \cdot \frac{(x+1)^3}{(x+2)^2} \right) \]
4Step 4: Simplify the Expression
Simplify the expression within the logarithm. This gives:\[ \log_{b} \left( \frac{x(x+1)^3}{(x+2)^2} \right) \]
Key Concepts
Power Rule of LogarithmsQuotient Rule of LogarithmsProduct Rule of Logarithms
Power Rule of Logarithms
The power rule of logarithms allows you to simplify expressions by moving constants as exponents inside the logarithmic argument. This can be a big help when dealing with complex expressions.
Here's how it works: If you have a term like \(n \log_{b}(A)\), where \(n\) is a constant, you can rewrite it as \(\log_{b}(A^n)\). This transformation can greatly simplify problems, especially when you're combining multiple logarithms.
In our exercise, the power rule is applied to both \(3 \log_{b}(x+1)\) and \(2 \log_{b}(x+2)\):
By converting the constants into exponents, we prepare the logarithmic terms for further combination using other logarithmic rules.
Here's how it works: If you have a term like \(n \log_{b}(A)\), where \(n\) is a constant, you can rewrite it as \(\log_{b}(A^n)\). This transformation can greatly simplify problems, especially when you're combining multiple logarithms.
In our exercise, the power rule is applied to both \(3 \log_{b}(x+1)\) and \(2 \log_{b}(x+2)\):
- \(3 \log_{b}(x+1) = \log_{b}((x+1)^3)\)
- \(2 \log_{b}(x+2) = \log_{b}((x+2)^2)\)
By converting the constants into exponents, we prepare the logarithmic terms for further combination using other logarithmic rules.
Quotient Rule of Logarithms
The quotient rule of logarithms is particularly handy when you're dealing with the subtraction of log expressions. It allows you to express the difference of two logarithms as the logarithm of a fraction.
This rule states: \(\log_{b}(A) - \log_{b}(B) = \log_{b}\left( \frac{A}{B} \right)\). The expression becomes cleaner and often simpler to manipulate further.
In the exercise, this rule is applied to \(\log_{b}((x+1)^3) - \log_{b}((x+2)^2)\), transforming it into:
This conversion simplifies the expression further by consolidating multiple logarithms into a single term, paving the way to incorporate more terms with ease.
This rule states: \(\log_{b}(A) - \log_{b}(B) = \log_{b}\left( \frac{A}{B} \right)\). The expression becomes cleaner and often simpler to manipulate further.
In the exercise, this rule is applied to \(\log_{b}((x+1)^3) - \log_{b}((x+2)^2)\), transforming it into:
- \(\log_{b} \left( \frac{(x+1)^3}{(x+2)^2} \right)\)
This conversion simplifies the expression further by consolidating multiple logarithms into a single term, paving the way to incorporate more terms with ease.
Product Rule of Logarithms
The product rule transforms the sum of two logarithms into a single logarithm comprising a product. This is useful when consolidating expressions into one simpler form.
The product rule states: \(\log_{b}(A) + \log_{b}(B) = \log_{b}(A \cdot B)\). By using this rule, you can easily combine log expressions.
In the exercise, after applying the quotient rule, we add \(\log_{b} x\) to our result:
With all the logarithmic expressions neatly wrapped into a single logarithm, the expression is both simpler and more elegant. The final equation \(\log_{b} \left( \frac{x(x+1)^3}{(x+2)^2} \right)\) effectively conveys the entirety of the original multiple terms within a unified and simplified form.
The product rule states: \(\log_{b}(A) + \log_{b}(B) = \log_{b}(A \cdot B)\). By using this rule, you can easily combine log expressions.
In the exercise, after applying the quotient rule, we add \(\log_{b} x\) to our result:
- \(\log_{b} \left( \frac{(x+1)^3}{(x+2)^2} \right) + \log_{b} x\) transforms into
\(\log_{b} \left( x \cdot \frac{(x+1)^3}{(x+2)^2} \right)\)
With all the logarithmic expressions neatly wrapped into a single logarithm, the expression is both simpler and more elegant. The final equation \(\log_{b} \left( \frac{x(x+1)^3}{(x+2)^2} \right)\) effectively conveys the entirety of the original multiple terms within a unified and simplified form.
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Problem 80
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