Problem 30 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt{e x} $$

Step-by-Step Solution

Answer

The expanded form of \(\ln \sqrt{e x}\) is \(1/2 + \ln x^{1/2}\).

Step 1: Simplify the square root

Rewrite the square root of \(ex\) as \(e^{1/2}x^{1/2}\). The square root of a number is equivalent to raising that number to the 1/2 power.

Step 2: Use properties of logarithms

Apply the properties of logarithms to expand the expression. According to the properties of logarithms, \(\ln(ab) = \ln a + \ln b\) and \(\ln a^n = n \ln a\), we can expand the expression into \(\ln e^{1/2} + \ln x^{1/2}\). Remember that the power rule of logarithms allows the exponent to be pulled to the front of the logarithm.

Step 3: Evaluate logarithmic expressions

Evaluate \(\ln e^{1/2}\). Because \(\ln e = 1\), this simplifies to \(1/2(1) = 1/2\). The final expanded form is \(1/2 + \ln x^{1/2}\).

Key Concepts

Logarithmic Expressions

When working with logarithmic expressions, it's like having a translator that helps us switch between two languages: exponential and logarithmic form. A logarithm basically asks the question, 'To what power do we need to raise a certain base to get a particular number?' For instance, the expression \(\ln x\) is asking, 'To what power do we raise \(e\), the base of the natural logarithm, to obtain \(x\)?'
Logarithmic expressions can be written in various forms and involve different operations, such as multiplication or division, which can be translated into addition or subtraction in the logarithmic world. Understanding how to correctly express and manipulate these expressions is crucial not only for expanding them but also for solving complex equations in higher-level mathematics.

Expanding Logarithms

Expanding logarithms is akin to untangling a knot, revealing each component of the logarithmic expression separately. To expand a logarithm means to take a complex expression, such as \(\ln(ab^c)\), and break it down into simpler parts, like \(\ln a + c\ln b\), using log rules. This process aids clarity and often simplifies the task of solving logarithmic equations. When we apply this to \(\ln \sqrt{ex}\), we untangle it into a clearer form, facilitating further operations or evaluations. Expanding provides insight into the structure of the expression, similar to how breaking a whole number into its prime factors can make arithmetic operations more straightforward.

Natural Logarithm

Imagine you've discovered a special tree in the forest of mathematics – this tree is known as the natural logarithm, represented by the symbol \(\ln\). The 'e' underneath is a mathematical constant approximately equal to 2.71828. The natural logarithm is the inverse operation of taking \(e\) to the power of a number and returns the exponent that \(e\) would be raised to get that number back. So \(\ln e\) is always equal to 1 because \(e\) raised to the power of 1 is just \(e\). This property makes the natural logarithm a fundamental tool in calculus and differential equations, appearing frequently in the natural sciences, economics, and engineering.

Logarithm Rules

Logarithm rules are the guidebook for navigating the landscape of logarithmic expressions. There are several key rules to master:

• The Product Rule: \(\ln(ab) = \ln a + \ln b\), which tells us how to deal with log of a product.
• The Quotient Rule: \(\ln\frac{a}{b} = \ln a - \ln b\), which explains logs of divisions.
• The Power Rule: \(\ln a^n = n\ln a\), letting us move the exponents in front of the log, simplifying computation.
• Change of Base Rule: Allows us to convert logs of one base to another, often to the common base of 10 or \(e\).

Utilizing these rules becomes second nature with practice, enabling us to simplify or solve logarithmic expressions with confidence. They are the steps to dance through any logarithmic problem with grace!