Problem 72
Question
Simplify each expression. $$10^{\log \sqrt[3]{x}}$$
Step-by-Step Solution
Verified Answer
The simplified form of the expression \(10^{\log \sqrt[3]{x}}\) is \(x^{1/3}\
1Step 1: Exponentiation Rule
Begin with the given expression: \(10^{\log \sqrt[3]{x}}\). The first step is to convert the cube root to exponential form. Recall that \(\sqrt[3]{x}\) can be written as \(x^{1/3}\), so rewrite the expression like this: \(10^{\log x^{1/3}}\)
2Step 2: Apply Logarithm Rule
Next, apply the power rule of logarithms, which allows the exponent of the argument of a logarithm to be brought out in front as a multiplier. The power rule states that \( \log_b(a^p) = p \log_b(a) \), so this gives us the expression \(10^{(1/3)*\log x}\
3Step 3: Simplification
Let's simplify further. Recall a fundamental property of exponents and logarithms: \(b^{(log_b(a))} = a\). Given that, \(10^{(1/3)*\log x}\) simplifies further to \((10^{\log x})^{1/3}\). Apply the property to simplify \(10^{\log x}\) to \(x\). So the expression simplifies to \(x^{1/3}\
Key Concepts
Logarithm RulesSimplifying ExpressionsCube Roots
Logarithm Rules
Logarithm rules are key mathematical properties that help simplify and solve equations involving logs. One such important property is the **power rule**.
This rule states that when you have a logarithmic expression of the form \( \log_b(a^p) \), it can be simplified to \( p \cdot \log_b(a) \).
This transformation is immensely useful for taking exponents out of a log expression, making the math easier to handle.
Another critical property is the **change of base formula**, which allows you to rewrite a logarithm in terms of logs of any base:
\( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \). This is particularly useful when solving problems that involve different bases.
Understanding these rules means you can efficiently tackle more complex logarithmic expressions by breaking them down into simpler components. Make sure to familiarize yourself with all fundamental log rules for greater ease in mathematical simplifications.
This rule states that when you have a logarithmic expression of the form \( \log_b(a^p) \), it can be simplified to \( p \cdot \log_b(a) \).
This transformation is immensely useful for taking exponents out of a log expression, making the math easier to handle.
Another critical property is the **change of base formula**, which allows you to rewrite a logarithm in terms of logs of any base:
\( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \). This is particularly useful when solving problems that involve different bases.
Understanding these rules means you can efficiently tackle more complex logarithmic expressions by breaking them down into simpler components. Make sure to familiarize yourself with all fundamental log rules for greater ease in mathematical simplifications.
Simplifying Expressions
Simplifying expressions is a crucial skill in mathematics that involves rewriting expressions in their most reduced or succinct form.
This can make them easier to evaluate or manipulate in equations.
Generally, simplifying involves combining like terms and using known mathematical properties to reduce the complexity of the expression.
For instance, in exponential expressions like \( 10^{\log x^{1/3}} \), using properties such as the logarithm rules helps to break down the expression step by step.
Here, we wield the properties:
This can make them easier to evaluate or manipulate in equations.
Generally, simplifying involves combining like terms and using known mathematical properties to reduce the complexity of the expression.
For instance, in exponential expressions like \( 10^{\log x^{1/3}} \), using properties such as the logarithm rules helps to break down the expression step by step.
Here, we wield the properties:
- The power rule for logarithms transforms the expression into \( (1/3) \cdot \log x \) form, making it easier to manage.
- Finally, knowing the link between exponents and logarithms, specifically \( b^{\log_b(a)} = a \), allows for further simplification.
Cube Roots
Cube roots refer to values that, when multiplied by themselves three times, return the original number.
The cube root of \( x \) is denoted as \( \sqrt[3]{x} \), or equivalently, \( x^{1/3} \).
Understanding cube roots is pivotal in many areas of algebra, as it relates to making roots part of ordinary algebraic manipulations.In practice, operations involving cube roots frequently appear in simplification exercises.
By expressing a cube root using its exponential form \( x^{1/3} \), it becomes easier to integrate it into expressions, such as when dealing with logarithms and exponents.
This represents a transition point in learning, moving from recognizing cube roots to using them effectively in more complex equations.Remember:
The cube root of \( x \) is denoted as \( \sqrt[3]{x} \), or equivalently, \( x^{1/3} \).
Understanding cube roots is pivotal in many areas of algebra, as it relates to making roots part of ordinary algebraic manipulations.In practice, operations involving cube roots frequently appear in simplification exercises.
By expressing a cube root using its exponential form \( x^{1/3} \), it becomes easier to integrate it into expressions, such as when dealing with logarithms and exponents.
This represents a transition point in learning, moving from recognizing cube roots to using them effectively in more complex equations.Remember:
- Cube roots are the inverse operation of raising a number to the power of three.
- Convert cube roots to their exponent form to facilitate simplification in expressions.
- Practicing converting between forms will enhance your ability to neatly solve algebraic problems.
Other exercises in this chapter
Problem 71
Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} 8$$
View solution Problem 71
Determine whether each statement "makes sense" or "does not make sense" and explair= your reasoning. My graph of \(f(x)=3 \cdot 2^{x}\) shows that the horizonta
View solution Problem 72
Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer.
View solution Problem 72
Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} 81$$
View solution