Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 3 through 9 , simplify the expression given. (a) \(10^{\frac{\log 8+1}{2}}\) (b) \(e^{-\frac{\ln 8}{3}+2}\)

Step-by-Step Solution

Answer

The simplified form of \(10^{\frac{\log 8+1}{2}}\) is \(√10 * 2^{\frac{3}{2}}\) and \(e^{-\frac{\ln 8}{3}+2}\) simplifies to \( \frac{e^2}{2}\).

Step 1: Simplify the Expression (a)

At first, rewrite \(10^{\frac{\log 8+1}{2}}\) as \(10^{\frac{\log 2^3+1}{2}}\). Now, using the rule of logarithms which states that log(a^b) = b*log(a), the expression can be simplified as \(10^{\frac{3*\log 2+1}{2}}\). Then, we rearrange to get \(10^{\frac{1+3*\log 2}{2}}\), which can be rewritten as \(10^{\frac{1}{2}+ \frac{3*\log 2}{2}}\) = \(10^{\frac{1}{2}} * 10^{\frac{3*\log 2}{2}}\) . The expression \(10^{\frac{3*\log 2}{2}}\) can also be simplified further using the relation \(10^{log x }= x\) as \(2^{\frac{3}{2}}\). Thus, the whole expression simplifies to \(√10 * 2^{\frac{3}{2}}\) .

Step 2: Simplify the Expression (b)

The expression \(e^{-\frac{\ln 8}{3}+2}\) can be rewritten as \(e^{-\frac{ln 2^3}{3}+2}\). Now, using similar principles of logarithm as before, the expression can be simplified as \(e^{-ln 2+2}\). Which can also be rewritten in another format like \(e^{2- ln 2}\), and further to \(e^{2} * e^{-ln 2}\). Note that \(e^{ln x }= x\), so this becomes \(e^2 * \frac{1}{2}\). Thus, the final simplified expression is \( \frac{e^2}{2}\).

Step 3: Final Answer

After simplifying the expressions, the final simplified form of \(10^{\frac{\log 8+1}{2}}\) is \( √10 * 2^{\frac{3}{2}}\) and of \(e^{-\frac{\ln 8}{3}+2}\) is \( \frac{e^2}{2}\).

Key Concepts

Logarithmic Identities

Logarithms are incredibly helpful tools in mathematics that allow us to simplify complex expressions, especially when dealing with exponents. A key identity to remember is the power rule, which states: \( \log(a^b) = b \cdot \log(a) \). This identity helps us break down an exponent into a more manageable form.
This application is seen in the solution where, for instance, the expression \( 10^{\frac{\log 2^3+1}{2}} \) is transformed using the power rule to simplify and separate the logarithmic part and constant part.

• Recognize the format: \( a^{\log_b c} = c^{\log_b a} \), this allows us to interchange bases.
• Utilize common logarithm simplifications: \( 10^{\log x} = x \) and \( e^{\ln x} = x \).

These identities not only simplify expressions but also aid in solving equations efficiently. Memorizing these rules can greatly enhance understanding and ability to work through complex problems quickly.

Exponential Functions

Exponential functions are functions where the variable is in the exponent, such as \( f(x) = a^x \). They appear frequently in scientific and financial calculations due to their rapid growth or decay properties. Understanding how to manipulate these functions is crucial in simplifying mathematical expressions.
In the solved exercise, the expressions \( 10^{\frac{1}{2}+ \frac{3\log 2}{2}} \) and \( e^{2- \ln 2} \) utilize the properties of exponential functions, allowing us to dissect them into simpler parts. For example:

• The rule \( a^{m+n} = a^m \cdot a^n \) allows us to break down individual components.
• Knowing that \( a^0 = 1 \) is useful for identifying simplificatory opportunities.

By comprehending these core principles, working with exponential expressions becomes less intimidating and more intuitive, making long strings of exponents manageable.

Expression Simplification

Expression simplification is a fundamental skill in algebra that makes complex calculations more straightforward. To simplify an expression means to reduce it to its simplest form without changing its value. This process often involves the use of mathematical identities and properties to rewrite the expression in a cleaner way.
In the given problem, breaking down parts like \( 10^{\frac{3 \log 2}{2}} \) to \( 2^{\frac{3}{2}} \) by recognizing that \( 10^{\log x} = x \) greatly aids in simplification.

• Divide and conquer: Tackle multiplication and division before addition and subtraction.
• Look for common factors or terms you can combine or reduce.

Expression simplification is not only about performing calculations but also about seeing patterns and utilizing them effectively. Over time, with practice, spotting these becomes intuitive.