Problem 32
Question
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log \sqrt[5]{\frac{x}{y}}\)
Step-by-Step Solution
Verified Answer
After we apply the rules of logarithm, the expression is expanded into \(\frac{1}{5} (\log x - \log y)\).
1Step 1: Identify the Root
The first step is to identify the root in the expression. It can be observed here that we have a fifth root, which implies the expression inside the root is raised to the power of \(1/5\). So, the expression can be written as \(\log (x/y)^{1/5}\).
2Step 2: Apply the Power Rule
Now apply the power rule which states that the logarithm of a number raised to a power is equal to the product of that power and the logarithm of that number. So, \(\log (x/y)^{1/5}\) can be rewritten as \(\frac{1}{5} \log (x/y)\).
3Step 3: Apply the Quotient Rule
The next step is to apply the quotient rule, which states that the logarithm of a quotient equals the difference of the logarithms of the numerator and denominator. The previous expression now becomes \(\frac{1}{5} (\log x - \log y)\).
Key Concepts
Power Rule for LogarithmsQuotient Rule for LogarithmsLogarithmic Expression Expansion
Power Rule for Logarithms
The power rule for logarithms is one of the fundamental properties that help simplify logarithmic expressions. This rule states that if you have a logarithm of a number raised to a power, you can bring that power down in front as a multiplier. In mathematical terms, it's expressed as:
The entire fraction \(x/y\) is raised to the power of \(1/5\).
Using the power rule, we simplify it by taking the exponent \(1/5\) in front of the logarithm.
So, it becomes \(\frac{1}{5} \cdot \log(x/y)\).
This transformation is helpful because it reduces the complexity of the logarithmic expression, making it easier to work with further.
- For a given base, \(\log_b(a^n) = n \cdot \log_b(a)\).
The entire fraction \(x/y\) is raised to the power of \(1/5\).
Using the power rule, we simplify it by taking the exponent \(1/5\) in front of the logarithm.
So, it becomes \(\frac{1}{5} \cdot \log(x/y)\).
This transformation is helpful because it reduces the complexity of the logarithmic expression, making it easier to work with further.
Quotient Rule for Logarithms
The quotient rule is essential in handling logarithms that involve division.
This rule simplifies a logarithm of a quotient into a subtraction problem, separating the log of the numerator and the log of the denominator.
In formulaic terms, it is described as:
The expression inside the logarithm function \(x/y\) splits into \(\log x - \log y\) due to the quotient property.
Thus, now the expression is \(\frac{1}{5} \cdot (\log x - \log y)\), which is a more expanded form and often makes evaluating or further manipulation much simpler.
This step is crucial to breaking down complex logs into simpler, smaller parts that can be easily tackled.
This rule simplifies a logarithm of a quotient into a subtraction problem, separating the log of the numerator and the log of the denominator.
In formulaic terms, it is described as:
- For the logarithm of a quotient, \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\).
The expression inside the logarithm function \(x/y\) splits into \(\log x - \log y\) due to the quotient property.
Thus, now the expression is \(\frac{1}{5} \cdot (\log x - \log y)\), which is a more expanded form and often makes evaluating or further manipulation much simpler.
This step is crucial to breaking down complex logs into simpler, smaller parts that can be easily tackled.
Logarithmic Expression Expansion
Expanding logarithmic expressions involves utilizing the properties of logarithms to rewrite the expression in a more detailed form.
In general, expansion helps in breaking down a compounded logarithmic expression into its individual logarithmic components.
The expansion process makes use of several rules, such as the power rule and the quotient rule, to achieve this.
Let's align this with our example: starting with \(\log \sqrt[5]{\frac{x}{y}}\), an expression containing a root and a quotient.
Consequently, expanded expressions are often easier to differentiate, integrate, or plug different values into, as needed for various calculations.
In general, expansion helps in breaking down a compounded logarithmic expression into its individual logarithmic components.
The expansion process makes use of several rules, such as the power rule and the quotient rule, to achieve this.
Let's align this with our example: starting with \(\log \sqrt[5]{\frac{x}{y}}\), an expression containing a root and a quotient.
- First, we identify that the root is \(1/5\) and bring it out in front of the logarithm using the power rule: \(\frac{1}{5} \cdot \log(x/y)\).
- Then, we utilize the quotient rule to further expand into \(\frac{1}{5}(\log x - \log y)\).
Consequently, expanded expressions are often easier to differentiate, integrate, or plug different values into, as needed for various calculations.
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