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Chapter 12: Problem 20

Express as an equivalent expression that is a product. $$\log _{10} x^{1 / 2}$$

Short Answer

Expert verified
\(\frac{1}{2} \cdot \log_{10}(x)\)

Step by step solution

01

Use the Power Rule for Logarithms

The power rule for logarithms states that \(\log_b(a^c) = c \cdot \log_b(a)\). Apply this rule to the given expression \(\log_{10}(x^{1/2})\).
02

Identify the Exponent

In the expression \(\log_{10}(x^{1/2})\), the exponent is \(1/2\).
03

Apply the Exponent to the Logarithm

Using the power rule, rewrite the expression as \(\frac{1}{2} \cdot \log_{10}(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule of Logarithms
The power rule of logarithms is a useful property that simplifies the process of working with logarithmic expressions involving exponents.
The rule states that if you have an expression in the form \(\text{log}_b(a^c)\), you can rewrite it as \(\text{log}_b(a^c) = c \cdot \text{log}_b(a)\).
This means you can 'pull out' the exponent as a coefficient in front of the logarithm.
For example, let's look at the original expression: \(\text{log}_{10}(x^{1/2})\).
According to the power rule, we can rewrite this expression as \(\frac{1}{2} \cdot \text{log}_{10}(x)\).
This transformation helps in simplifying complex logarithmic equations and makes them easier to work with.
Exponents
Exponents represent the power to which a number or expression is raised. They are crucial in both algebra and logarithmic calculations.
In our given expression \( \text{log}_{10}(x^{1/2}) \), the exponent is \(1/2\).
Exponents can also be shown as fractions, which are common in square roots, cube roots, etc.
For example:
  • \(x^{1/2} = \sqrt{x}\) (square root)
  • \(x^{1/3} = \sqrt[3]{x}\) (cube root)
Using exponents in logarithms often involves applying properties like the power rule, making complex expressions simpler.
Logarithmic Expressions
Logarithmic expressions involve the logarithm function, which is the inverse of exponentiation.
Logarithms have several properties that make manipulating these expressions easier.
For the given exercises involving \( \text{log}_{10} \), the base of the logarithm is 10 (known as the common logarithm).
Some useful properties of logarithms include:
  • Product Rule: \(\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\)
  • Quotient Rule: \(\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)\)
  • Power Rule: \(\text{log}_b(a^c) = c \cdot \text{log}_b(a)\)
These properties allow for the simplification and manipulation of more complex logarithmic expressions, turning them into more manageable forms for further calculations.

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Most popular questions from this chapter

Find the domain of each function. $$f(x)=\frac{x-3}{x+7}$$

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Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} \frac{3}{5}$$
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