Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each expression. $$\ln e^{9 x}$$

Step-by-Step Solution

Answer

The simplified expression of \(\ln(e^{9x})\) is \(9x\).

Step 1: Apply Logarithm Laws

Simplify the expression \(\ln(e^{9x})\) using the law of logarithms which states that \(\ln(a^b) = b \ln(a)\). The expression, therefore, simplifies to \(9x \ln(e)\).

Step 2: Simplify Further

Recall that the \(\ln(e) = 1\), because e raised to the first power yields e. Thus, we get \(9x \cdot 1\), which gives \(9x\).

Key Concepts

Logarithm Laws

Understanding logarithm laws is crucial when simplifying logarithmic expressions. Logarithms are the inverses of exponential functions, meaning they undo the exponential process. For instance, the equation \(\log_b a = c\) signifies that \(b^c = a\). Logarithm laws, or properties, allow us to manipulate and simplify complex logarithmic expressions into more manageable forms.

Here are some essential logarithm laws:

• The Product Rule: \(\log(a \cdot b) = \log(a) + \log(b)\), which allows us to turn the logarithm of a product into a sum of logarithms.
• The Quotient Rule: \(\log(\frac{a}{b}) = \log(a) - \log(b)\), which transforms the logarithm of a quotient into a difference of logarithms.
• The Power Rule: \(\log(a^b) = b \cdot \log(a)\), enabling us to bring the exponent in front of the log.
• Change of Base Formula: \(\log_b a = \frac{\log_c a}{\log_c b}\), useful for converting logs of different bases.

These properties become particularly helpful when working with natural logarithms, where the base of the logarithm is the special number \(e\). In the given exercise, the Power Rule simplified \(\ln(e^{9x})\) to \(9x \(\ln(e)\)\).

Natural Logarithm Properties

The natural logarithm, denoted as \(\ln(x)\), is a logarithm with base \(e\), where \(e\) is approximately equal to 2.71828. This irrational and transcendental number is the base of the natural exponential function and plays a pivotal role in continuous growth models. Properties of natural logarithms are specialized versions of the general logarithm laws, tailored for base \(e\).

Notable properties of natural logarithms include:

• Identity Property: \(\ln(e) = 1\), because \(e\) to the power of 1 is \(e\). This property was used in the solution to simplify \(9x \(\ln(e)\)\) to just \(9x\).
• Inversion Property: \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\), reflecting the inverse relationship between \(e\) and \(\ln\).
• Log of One: \(\ln(1) = 0\), since \(e^0 = 1\).
• Log of a Product: \(\ln(a \cdot b) = \ln(a) + \ln(b)\).
• Log of a Quotient: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\).
• Log of a Power: \(\ln(a^b) = b \(\ln(a)\)\), which is an instance of the general Power Rule.

These properties enable clear and concise simplification of expressions, and understanding them aids in recognizing how to manipulate natural logarithm equations effectively.

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. The general form of an exponential function is \(f(x) = a^{bx}\), where the base \(a\) is a positive real number, and \(b\) is the rate or exponent that scales the variable \(x\).

The natural exponential function has the form \(f(x) = e^x\), with the base as the Euler's number \(e\). It's one of the most important functions in mathematics due to its unique properties, being its own derivative and integral. This feature makes \(e^x\) extremely significant in calculus, particularly in solving differential equations that model real-world phenomena such as population growth, radioactive decay, and compound interest.

In regards to the exercise, \(\ln(e^{9x})\) involves the natural exponential function where \(e\) is raised to \(9x\). The property that \(\ln(e^x) = x\) is essential here, because it utilizes the inverse nature of logarithmic and exponential functions. Overall, exponential functions and their properties are indispensable tools for simplifying and understanding complex problems in mathematics and other scientific disciplines.