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Magazine Chapter 12: Problem 30
Find the exact value of each logarithm. $$ \log \sqrt{10} $$
Short Answer
Expert verified
The exact value is \( \frac{1}{2} \).
Step by step solution
01
Understanding the Problem
We need to find the exact value of the logarithm \( \log \sqrt{10} \). This means finding the power to which the base (implicitly 10 when not specified) must be raised to get \( \sqrt{10} \).
02
Rewriting the Expression
The expression \( \sqrt{10} \) can be rewritten using fractional exponents. Thus, \( \sqrt{10} = 10^{1/2} \).
03
Using Logarithm Power Rule
Utilize the power rule of logarithms: \( \log(a^b) = b \cdot \log(a) \). Applying it here:\[ \log(10^{1/2}) = \frac{1}{2} \cdot \log(10) \]
04
Simplifying Using Logarithm Identity
We know that \( \log(10) = 1 \) because 10 is the base of the common logarithms (base-10). Substitute this value into the expression:\[ \frac{1}{2} \cdot 1 = \frac{1}{2} \]
05
Final Answer
Thus, the exact value of \( \log \sqrt{10} \) is \( \frac{1}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithm Power Rule
The logarithm power rule is a very useful tool when solving logarithmic expressions. It states: \( \log(a^b) = b \cdot \log(a) \). This rule helps simplify logarithms that contain exponents. By using this rule, you can pull the exponent in the logarithmic function out as a multiple.
This process makes calculations easier and more straightforward.
This process makes calculations easier and more straightforward.
- If you have a logarithm, such as \( \log(10^{1/2}) \), you can apply the power rule to simplify it to \( \frac{1}{2} \cdot \log(10) \).
- This extraction makes it clear how exponents manipulate logarithmic values.
Fractional Exponents
Fractional exponents are a way of expressing roots in a different form. Instead of writing roots like \( \sqrt{10} \), you can express the same operation with exponents: \( 10^{1/2} \). This notation helps in simplifying calculations, especially when dealing with logarithms.
- For example, the square root of a number, \( \sqrt{x} \), can be rewritten as \( x^{1/2} \).
- Likewise, a cube root, or \( \sqrt[3]{x} \), becomes \( x^{1/3} \).
Common Logarithms
Common logarithms are logarithms with base 10. They are frequently used in mathematics, science, and engineering because our number system is base 10. When you see a logarithm without an explicitly written base, it's usually a common logarithm. The notation "log" without a subscript implies base 10.
- For instance, \( \log(100) \) asks "to what power must 10 be raised to result in 100?" The answer is \( 2 \), since \( 10^2 = 100 \).
- Common logarithms make calculations intuitive when working with powers of 10 since humans commonly use these numbers.
Logarithm Identity
A crucial identity in logarithms is that \( \log_{10}(10) = 1 \). This identity states that any number raised to the power of its own base gives 1 when expressed in logarithms. In the context of common logarithms: \( \log(10) = 1 \).
Such an identity is a foundation for simplifying more complex expressions.
Such an identity is a foundation for simplifying more complex expressions.
- For example, if you have an expression like \( \frac{1}{2} \cdot \log(10) \), knowing \( \log(10) = 1 \) allows you to quickly simplify it to \( \frac{1}{2} \).
- This helps in breaking down and verifying results efficiently.
