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Chapter 4: Problem 10

Simplify the following expressions. \(e^{\ln x^{2}+3 \ln y}\)

Short Answer

Expert verified
The simplified expression is \( x^2 y^3 \).

Step by step solution

01

Use the properties of logarithms

The expression inside the exponent is \( \ln{x^2} + 3 \ln{y} \). Using the properties of logarithms, specifically \( \ln{a^b} = b \ln{a} \) and the fact that \( \ln{a} + \ln{b} = \ln{(ab)} \), we can rewrite it as follows: \( \ln{x^2} + 3 \ln{y} = 2 \ln{x} + \ln{y^3} \).
02

Combine logarithmic terms

Combine the logarithmic terms: \( 2 \ln{x} + 3 \ln{y} = \ln{x^2} + \ln{y^3} \). Using the logarithm property \( \ln{a} + \ln{b} = \ln{(ab)} \), we can combine them: \( \ln{x^2} + \ln{y^3} = \ln{x^2 y^3} \).
03

Apply the exponential function

Now replace back into the original expression: \( e^{\left( \ln{x^2 y^3}\right)} \). Using the property that \( e^{\ln{a}} = a \), we simplify it to: \( x^2 y^3 \).

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Key Concepts

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

logarithms
Logarithms are a mathematical concept used to solve exponential equations. They are the inverse operation of exponentiation. For example, if we have a number \(a\) raised to the power of \(b\) to get \(c \), written as \(a^b = c\), the logarithm helps us solve for \(b\) by using \(\log_a{(c)} = b\). Logs have critical properties that make them very useful in calculus:
  • \textbf{Product Rule:} \[ \log_b{(xy)} = \log_b{(x)} + \log_b{(y)} \]. It states that the log of a product is the sum of the logs of the factors.
  • \textbf{Quotient Rule:} \[ \log_b{\left( \frac{x}{y} \right)} = \log_b{(x)} - \log_b{(y)} \]. This indicates that the log of a quotient is the difference of the logs.
  • \textbf{Power Rule:} \[ \log_b{(x^y)} = y \log_b{(x)} \]. This rule shows that the log of a number raised to a power can be simplified by multiplying the power by the log.
These properties are essential when simplifying log expressions.
exponential functions
Exponential functions involve expressions where a constant base is raised to a variable exponent, like \(a^x\). In calculus, the base 'e' (approximately 2.71828) is particularly important, as it is the base of the natural logarithm. The exponential function \(e^x\) has unique properties:
  • \textbf{Continuity and Differentiability:} \(e^x\) is continuous and differentiable everywhere.
  • \textbf{Derivative:} The derivative of \(e^x\) is itself, \(\frac{d}{dx}{e^x} = e^x\). This is a vital property in calculus.
  • \textbf{Inverse:} The inverse function of \(e^x\) is the natural logarithm, \(\ln(x)\). This means \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\).
Exponential functions grow rapidly and are used to model many real-world processes, such as population growth, radioactive decay, and interest compounding.
simplifying expressions
Simplifying expressions in calculus often involves combining like terms, factoring, or using algebraic identities. Here are some key steps to keep in mind:
  • \textbf{Identify Like Terms:} Look for terms that can be combined. For instance, \(2\ln{x} + 3\ln{y}\) can be approached using the properties of logarithms to combine these terms.
  • \textbf{Use Algebraic Rules:} Using rules such as the logarithmic properties mentioned earlier helps simplify expressions. For instance, \(\ln{x^2} + 3\ln{y}\) becomes \(2\ln{x} + \ln{y^3}\) and further simplifies to \(\ln{x^2y^3}\).
  • \textbf{Substitution:} Replace simplified parts of the expression into the original form. For example, converting \(e^{\ln{x^2y^3}}\) using the inverse property \(e^{\ln{a}} = a\) simplifies to \(x^2 y^3\).
Simplifying expressions is a crucial skill in calculus to make solving problems more manageable.

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