Problem 131 - College Algebra

Question

Write as a single term that does not contain a logarithm: \(e^{\ln 8 x^{3}-\ln 2 x^{2}}\)

Step-by-Step Solution

Answer

The simplified form of \(e^{\ln 8 x^{3}-\ln 2 x^{2}}\) is \(4x\).

Step 1: Simplify the Expression inside the Exponent

First, apply the properties of logarithms inside the exponent. According to the log subtraction rule, \(\ln a - \ln b = \ln \frac{a}{b}\). So, \(\ln 8 x^{3}-\ln 2 x^{2}\) can be rewritten as \(\ln \frac{8 x^{3}}{2 x^{2}}\).

Step 2: Further Simplify the Expression

Simplify the fraction inside the logarithm to get \(\ln 4x\). The fraction \(\frac{8 x^{3}}{2 x^{2}}\) simplifies to \(4x\). So, the original expression now becomes \(e^{\ln 4x}\).

Step 3: Simplify using the Relationship between \(e\) and \(\ln\)

The final step is to use the property that \(e^{\ln a} = a\). Therefore, \(e^{\ln 4x}\) simplifies to \(4x\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying logarithmic expressions and solving logarithmic equations. These properties stem from the basic definition of logarithms as the inverses of exponential functions.

One key property used in simplifying expressions like the exercise at hand is the quotient rule, which states that the logarithm of a quotient is equal to the difference of the logarithms, \textbf{i.e.}, \( \log(\frac{a}{b}) = \log(a) - \log(b) \). This makes it possible to turn a potentially complicated expression into a simpler one by breaking down larger components into their constituent parts.

Additionally, the product rule \( \log(ab) = \log(a) + \log(b) \) and the power rule \( \log(a^b) = b\log(a) \) are also widely used. These rules allow us to manipulate logarithmic expressions to a form in which they can be more easily computed or further simplified, as shown in the exercise.

Simplifying Logarithms

When simplifying logarithms, applying the properties of logarithms properly ensures an expression is reduced to its simplest form. Exercises such as the one provided demonstrate how logarithms involving algebraic expressions can often be broken down and reconstructed into a more manageable form.

Starting with the understanding that certain logarithmic statements can be combined or separated based on their operations (multiplication turning into addition, division into subtraction, etc.), we can restructure expressions to isolate variables or numbers. This process often reveals a clearer path towards solving equations or simplifying terms, moving us closer to an expression without the need for a logarithm, as seen in the given sample solution.

By using these properties effectively, complex expressions can be unraveled step-by-step, eventually reducing them to their simplest algebraic or numeric equivalent.

Exponential and Logarithmic Relationship

The exponential and logarithmic relationship is at the heart of understanding how to work with logarithms and exponential functions interchangeably. Because logarithms are defined as the inverse of exponential functions, we have an extremely important identity: \( e^{\log(a)} = a \) when referring to the natural logarithm \( \log \) as \( \ln \). This relationship played a central role in our exercise, bringing us from a logarithmic expression within an exponent to a simplified algebraic term.

In the context of the exercise, this means we can reverse the operation of taking the natural logarithm (\( \ln \) of a number by raising \( e \) to the power of that logarithm — effectively 'canceling' out the log function and leaving us with the original number or variable in question. This identity is one of the simplest yet most profound within mathematics, enabling learners to transition between expressions and their logarithmic equivalent with ease.