Question

Use the properties of logarithms to expand the logarithmic expression. $$ \ln \sqrt{2^{3}} $$

Short answer

\(\frac{3}{2}*\ln(2)\)

Step-by-step solution

Identify the property

The logarithmic expression involves a value raised to a power inside the square root. This allows us to use two properties of logarithms: \(\ln(a^b) = b \cdot \ln(a)\) and \(\ln(\sqrt{a}) = \frac{1}{2} \ln(a)\).

Use the property \(\ln(a^b) = b \cdot \ln(a)\)

The logarithmic expression can be rewritten as \(\ln((2^{3})^{1/2})\). Using the property outlined in step 1, this can be simplified to \(\frac{1}{2}*3*\ln(2)\).

Simplify the expression

Lastly, simplify the expression to get \(\frac{3}{2}*\ln(2)\).

Key concepts

Logarithmic Expressions

Understanding logarithms is essential for working with exponentials and modeling real-world situations, from measuring earthquakes with the Richter scale to describing the loudness of sound in decibels.

Logarithmic expressions involve the log function, which is the inverse of exponentiation. In other words, if you have an equation like \( a^b = c \), the logarithm tells you what exponent b you need to raise a to get c. The expression \( \log_{a}(c) \) is the exponent b. In the world of logarithms, there are a few key properties that help simplify expressions:

• The Product Rule: \( \log_{a}(bc) = \log_{a}(b) + \log_{a}(c) \)
• The Quotient Rule: \( \log_{a}(\frac{b}{c}) = \log_{a}(b) - \log_{a}(c) \)
• The Power Rule: \( \log_{a}(b^c) = c \cdot \log_{a}(b) \)
• Change of Base Formula: \( \log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)} \), where c is any positive value.

Using these properties can greatly simplify complex logarithmic expressions and make them more manageable to work with.

Natural Logarithm

The natural logarithm is a special type of logarithmic expression where the base is the mathematical constant \( e \), roughly equal to 2.71828. This constant is used throughout mathematics, especially in calculus and complex analysis, because of its unique properties when it comes to rates of growth.

The natural logarithm is denoted by \( \ln \). It might seem just like a symbol, but it has profound implications in various fields including physics, finance, and engineering, among others, to describe continuous growth or decay. For example, in finance, the natural logarithm is used to calculate continuous compounding interest.

An important property of the natural logarithm is that \( \ln(e) = 1 \) because \( e^1 = e \). When dealing with natural logarithms, you can also apply the same rules as for common logarithms, such as the power rule which was used in the exercise to simplify \( \ln \sqrt{2^{3}} \) to \( \frac{3}{2} * \ln(2) \).

Exponential Functions

Exponential functions are intimately tied with logarithms; understanding one means understanding the other. An exponential function is of the form \( y = a^x \), where a is a positive constant. It represents growth or decay that increases at a rate proportional to the current amount.

The number e, the base of the natural logarithm, is what you often see as the base in 'natural' exponential functions, written as \( y = e^x \). These functions are crucial in modeling situations where things grow rapidly and continuously, such as populations, radioactive decay, or investment growth.

One of the keys to working with exponential functions is understanding their inverse relationship with logarithms. While an exponential function tells you the output given a power, a logarithm tells you the power given the output. They are two sides of the same coin, and knowledge of one enhances understanding of the other. In practice, recognizing this inverse relationship is vital in solving equations involving exponentials and logarithms and is exemplified in the exercise where properties of logarithms are applied to an exponential expression within a natural logarithm.