Problem 20 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$-19 e^{-2 \ln x^{2}}$$

Step-by-Step Solution

Answer

\(-19 x^{-4}\)

Step 1: Understanding the expression

The given expression is \(-19 e^{-2 \ln x^{2}}\). We need to focus on the term \(e^{-2 \ln x^{2}}\) to simplify it using the properties of logarithms.

Step 2: Apply the power rule of logarithms

Rewrite the logarithmic term \(\ln x^{2}\) using the power rule: \(\ln x^{2} = 2 \ln x\). This simplifies the expression inside the exponent to \(-2 \ln x^{2} = -2(2 \ln x) = -4 \ln x\).

Step 3: Apply the property of exponents and logarithms

Using the property \(e^{a \ln b} = b^{a}\), we can simplify \(e^{-4 \ln x}\) to \(x^{-4}\) by letting \(a = -4\) and \(b = x\).

Step 4: Simplify the entire expression

Substitute the simplified term back into the original expression:\[-19 e^{-2 \ln x^{2}} = -19 x^{-4}.\]This is the simplified form of the given expression, applying the properties of logarithms and exponents.

Key Concepts

Properties of Logarithms

Logarithms come with a set of properties that make them particularly useful in simplifying complex mathematical expressions. These properties allow us to transform and manipulate logarithmic expressions, making them easier to work with.
One very handy property of logarithms is the "product rule," which states that the logarithm of a product is the sum of the logarithms of its factors. Mathematically, this can be expressed as: \( \log_b(mn) = \log_b(m) + \log_b(n) \).
Another essential property is the "quotient rule," which shows that the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator: \( \log_b(m/n) = \log_b(m) - \log_b(n) \).
Most relevant to our discussion is the "power rule" of logarithms. This rule states that the logarithm of an expression raised to a power can be simplified by bringing the power in front of the logarithm: \( \log_b(m^n) = n \log_b(m) \).
These properties are tools that help in simplifying expressions, especially when working with variables and exponents.

Power Rule of Logarithms

The power rule is a powerful tool when dealing with logarithms. Simplifying expressions like the one in our exercise heavily relies on this rule.
When you see a logarithmic term like \( \ln x^2 \), the power rule allows you to take the exponent 2 and bring it in front of the logarithm: \( \ln x^2 = 2 \ln x \). This simplifies the expression, making it easier to handle.
Using the power rule can be particularly helpful in expressions that involve exponential and logarithmic forms. It allows you to transform the expression into a format that is easier to manipulate. For example, in the term \( -2 \ln x^2 \), applying the rule gives \( -4 \ln x \), simplifying the problem significantly.
It's like opening a locked door; once you know how to use it, the rest of the expression often simplifies nicely with a bit of algebra.

Exponents and Logarithms Relationship

Understanding the relationship between exponents and logarithms is key to mastering exercises involving these concepts.
A fundamental property of exponents and logarithms is that they are inverse operations. This means applying a logarithm and then an exponential function, or vice versa, can simplify expressions significantly.
Particularly useful is the property \( e^{a \ln b} = b^a \). This property tells us that an exponential term with a logarithm can be directly converted into a power form. For instance, \( e^{-4 \ln x} \) can be easily rewritten as \( x^{-4} \) using this property.
This transition from exponential-logarithmic forms to power forms makes solving equations and simplifying expressions more straightforward. It allows for a seamless move from seemingly complex logarithmic and exponential expressions to simpler algebraic forms.
Grasping how these operations interplay provides a strong foundation for tackling more advanced problems in calculus and beyond.