Problem 24
Question
Use the Laws of Logarithms to expand the expression. $$\ln \sqrt{z}$$
Step-by-Step Solution
Verified Answer
The expanded expression is \(\frac{1}{2} \ln(z)\).
1Step 1: Express the Square Root as Exponentiation
Start by writing the square root of a variable as an exponent. The square root of \(z\) is the same as \(z^{1/2}\). So, rewrite the expression \(\ln \sqrt{z}\) as \(\ln(z^{1/2})\).
2Step 2: Apply the Power Rule for Logarithms
The Power Rule of Logarithms states that for any logarithm \(\log_b(a^c) = c \cdot \log_b(a)\). Using this rule, \(\ln(z^{1/2})\) becomes \(\frac{1}{2} \ln(z)\).
3Step 3: Conclusion: Simplified Expression
The initial expression \(\ln \sqrt{z}\) has been expanded. The final expression is \(\frac{1}{2} \ln(z)\).
Key Concepts
Logarithmic FunctionsExponentiationSquare RootPower Rule
Logarithmic Functions
Logarithms are mathematical operations that unravel the mysteries behind exponential functions. They express how many times one number, called the base, must be multiplied by itself to reach another number. In simpler terms, if you have a number like 8 and want to know how many times you must multiply 2 to get it, logarithms can tell you.
Logarithmic functions are the reverse of exponential functions and are key in solving equations involving exponents. They are written as \( \log_b(a) \), where \( b \) is the base and \( a \) is the result of \( b \) raised to some power. Natural logarithms use the constant \( e \) as their base and are often denoted as \( \ln \).
In the exercise "\( \ln \sqrt{z} \)," the logarithm used is natural, and it's important to know the properties of logarithms to expand or simplify expressions.
Logarithmic functions are the reverse of exponential functions and are key in solving equations involving exponents. They are written as \( \log_b(a) \), where \( b \) is the base and \( a \) is the result of \( b \) raised to some power. Natural logarithms use the constant \( e \) as their base and are often denoted as \( \ln \).
In the exercise "\( \ln \sqrt{z} \)," the logarithm used is natural, and it's important to know the properties of logarithms to expand or simplify expressions.
Exponentiation
Exponentiation is the process of raising a number to a power. The result tells you how many times to multiply the number by itself. For example, \( 2^3 \) means 2 multiplied by itself three times, which equals 8.
When dealing with exponential expressions, we're often interested in how to manipulate and simplify them, particularly when they appear within logarithmic problems.
In the context of logarithmic expressions, exponentiation is used to express square roots and expand logarithmic forms. For example, the square root of \( z \) is \( z^{1/2} \), as seen in the transformation of \( \ln \sqrt{z} \) into \( \ln(z^{1/2}) \). This simplification is foundational in making logarithmic expressions easier to work with.
When dealing with exponential expressions, we're often interested in how to manipulate and simplify them, particularly when they appear within logarithmic problems.
In the context of logarithmic expressions, exponentiation is used to express square roots and expand logarithmic forms. For example, the square root of \( z \) is \( z^{1/2} \), as seen in the transformation of \( \ln \sqrt{z} \) into \( \ln(z^{1/2}) \). This simplification is foundational in making logarithmic expressions easier to work with.
Square Root
A square root of a number is a value that, when multiplied by itself, gives the original number. If \( z^2 = y \), then \( z \) is the square root of \( y \), noted as \( \sqrt{y} \).
Square roots are common in various mathematical problems and are fundamental in algebra and calculus. When square roots appear in logarithmic functions, like in \( \ln \sqrt{z} \), it's often useful to express them in terms of exponentiation.
This method converts the square root to a fractional exponent, turning \( \sqrt{z} \) into \( z^{1/2} \). Rewriting square roots in this way allows for the application of logarithmic properties, like the power rule, to simplify complex expressions.
Square roots are common in various mathematical problems and are fundamental in algebra and calculus. When square roots appear in logarithmic functions, like in \( \ln \sqrt{z} \), it's often useful to express them in terms of exponentiation.
This method converts the square root to a fractional exponent, turning \( \sqrt{z} \) into \( z^{1/2} \). Rewriting square roots in this way allows for the application of logarithmic properties, like the power rule, to simplify complex expressions.
Power Rule
The power rule in logarithms is a handy tool, especially when you want to simplify expressions involving exponents. This rule states that \( \log_b(a^c) = c \times \log_b(a) \).
Simply put, it allows you to bring the exponent in front of the logarithm, turning multiplication inside the log into a simpler arithmetic operation. This rule is central to solving complex logarithmic functions, as it can break down seemingly complicated expressions into manageable parts.
In the example of expanding \( \ln \sqrt{z} \), we first rewrite it as \( \ln(z^{1/2}) \). Then, by applying the power rule, we convert it to \( \frac{1}{2} \ln(z) \). This transformation exemplifies the utility of the power rule, making the expression clearer and easier to evaluate.
Simply put, it allows you to bring the exponent in front of the logarithm, turning multiplication inside the log into a simpler arithmetic operation. This rule is central to solving complex logarithmic functions, as it can break down seemingly complicated expressions into manageable parts.
In the example of expanding \( \ln \sqrt{z} \), we first rewrite it as \( \ln(z^{1/2}) \). Then, by applying the power rule, we convert it to \( \frac{1}{2} \ln(z) \). This transformation exemplifies the utility of the power rule, making the expression clearer and easier to evaluate.
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