Problem 47
Question
Write each expression in terms of \(A\) and \(B\) if \(\log _{2} x=A\) and \(\log _{2} y=B\). \(\log _{2} \sqrt[4]{y}\)
Step-by-Step Solution
Verified Answer
\(\log_{2} \sqrt[4]{y} = \frac{1}{4}B\).
1Step 1: Understand the Expression
First, recognize that the expression \( \log_{2} \sqrt[4]{y} \) involves a logarithm and a root operation. The root can be rewritten as a power for easier manipulation in logarithmic terms.
2Step 2: Rewrite as a Power
Express the fourth root of \( y \) in terms of an exponent: \( \sqrt[4]{y} = y^{1/4} \). This will help us apply logarithmic rules more effectively.
3Step 3: Apply Logarithmic Power Rule
Utilize the power rule of logarithms, which states \( \log_{b}(m^n) = n \cdot \log_{b}(m) \), to rewrite the logarithmic expression as \( \log_{2}(y^{1/4}) = \frac{1}{4} \cdot \log_{2}(y) \).
4Step 4: Substitute Known Logarithmic Value
Substitute \( \log_{2} y = B \) into the expression: \( \frac{1}{4} \cdot \log_{2} y = \frac{1}{4} \cdot B \).
5Step 5: Simplify
The expression simplifies to \( \frac{1}{4}B \), which is the logarithmic expression in terms of \( B \).
Key Concepts
Logarithmic operationsExponents and rootsProperties of logarithms
Logarithmic operations
In mathematics, logarithms are an essential concept used to transform multiplicative relationships into additive ones. This simplification is beneficial in solving equations and analyzing exponential growth. A logarithmic operation involves applying the logarithmic function, which is the inverse of an exponentiation. In simple terms, given a base, a logarithm tells us the power to which the base must be raised to yield a particular number.
For example, if you see an expression like \( \log_{2} x \), it asks, "To what power must 2 be raised to result in \( x \)?" This operation is not only used in solving mathematical problems but is also crucial in many fields such as science and engineering. Understanding how to handle and manipulate logarithmic expressions, such as using logarithmic rules (like the power, product, and quotient rules), simplifies complex problems significantly. These rules help rewrite logarithms, making them more manageable to work with in various calculations.
For example, if you see an expression like \( \log_{2} x \), it asks, "To what power must 2 be raised to result in \( x \)?" This operation is not only used in solving mathematical problems but is also crucial in many fields such as science and engineering. Understanding how to handle and manipulate logarithmic expressions, such as using logarithmic rules (like the power, product, and quotient rules), simplifies complex problems significantly. These rules help rewrite logarithms, making them more manageable to work with in various calculations.
- The Power Rule: \( \log_{b}(m^n) = n \cdot \log_{b}(m) \)
- The Product Rule: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \)
- The Quotient Rule: \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \)
Exponents and roots
Exponents and roots are fundamental concepts in mathematics that describe how numbers are raised to a power (exponents) or how roots are extracted from them. Understanding these concepts is crucial for working through logarithmic expressions effectively.
Exponents, denoted by a superscript, show how many times a number, known as the base, is multiplied by itself. For instance, \( y^4 \) means \( y \times y \times y \times y \). When dealing with roots, like square roots, cube roots, and fourth roots, you're essentially asking, "What number can be multiplied by itself a certain number of times to give the original number?"
In our exercise, the fourth root of \( y \), represented as \( \sqrt[4]{y} \), can be expressed as an exponent: \( y^{1/4} \). This transformation is pivotal because once roots are expressed as powers, logarithmic rules like the power rule can seamlessly be applied.
Exponents, denoted by a superscript, show how many times a number, known as the base, is multiplied by itself. For instance, \( y^4 \) means \( y \times y \times y \times y \). When dealing with roots, like square roots, cube roots, and fourth roots, you're essentially asking, "What number can be multiplied by itself a certain number of times to give the original number?"
In our exercise, the fourth root of \( y \), represented as \( \sqrt[4]{y} \), can be expressed as an exponent: \( y^{1/4} \). This transformation is pivotal because once roots are expressed as powers, logarithmic rules like the power rule can seamlessly be applied.
- Square root: \( \sqrt{y} = y^{1/2} \)
- Fourth root: \( \sqrt[4]{y} = y^{1/4} \)
- General root \( n \): \( \sqrt[n]{y} = y^{1/n} \)
Properties of logarithms
The properties of logarithms are rules that provide guidelines on how to simplify and manipulate logarithmic expressions. These properties help in breaking down complex logarithmic equations into simpler parts, making it easier to find solutions or make specific substitutions.
One key property used in the original exercise is the Power Rule of Logarithms, which states that \( \log_{b}(m^n) = n \cdot \log_{b}(m) \). This rule allows you to move the exponent in front of the log, turning a multiplication inside the log into a simple multiplication outside.
For instance, in the expression \( \log_{2}(y^{1/4}) \), by applying the power rule, it can be rewritten as \( \frac{1}{4} \cdot \log_{2}(y) \). This showcases how properties of logarithms can transform an equation by breaking down or reorganizing components, which can be especially useful for further calculations or simplifications.
To successfully apply these properties, it's important to be well-versed with each rule and recognize opportunities to use them in expressions:
One key property used in the original exercise is the Power Rule of Logarithms, which states that \( \log_{b}(m^n) = n \cdot \log_{b}(m) \). This rule allows you to move the exponent in front of the log, turning a multiplication inside the log into a simple multiplication outside.
For instance, in the expression \( \log_{2}(y^{1/4}) \), by applying the power rule, it can be rewritten as \( \frac{1}{4} \cdot \log_{2}(y) \). This showcases how properties of logarithms can transform an equation by breaking down or reorganizing components, which can be especially useful for further calculations or simplifications.
To successfully apply these properties, it's important to be well-versed with each rule and recognize opportunities to use them in expressions:
- The power property, as shown: \( \log_{b}(m^n) = n \cdot \log_{b}(m) \)
- The product property: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \)
- The quotient property: \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \)
Other exercises in this chapter
Problem 47
In \(45-52,\) if \(\ln a=c,\) express each of the following in terms of \(c\) $$ \ln a^{-1} $$
View solution Problem 47
\(\operatorname{In} 36-47,\) if \(\log 3=x\) and \(\log 5=y,\) write each of the logs in terms of \(x\) and \(y\) $$ \log \left(\frac{3}{5}\right)^{2} $$
View solution Problem 48
In \(45-52,\) if \(\ln a=c,\) express each of the following in terms of \(c\) $$ \ln \frac{1}{a} $$
View solution Problem 48
In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log a^{2} $$
View solution