Problem 37 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58,\) use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln 4 x$$

Step-by-Step Solution

Answer

After expansion, the expression is \(\ln 4 + \ln x\).

Step 1: Identifying logarithmic properties to use

In this problem, the property of logarithms that says \(ln(ab) = ln(a) + ln(b)\) is relevant to separate the multiplication inside the logarithms into separate log terms.

Step 2: Apply the property of logarithms

Apply the mentioned property on the expression \(\ln 4x\). This results in \(\ln 4 + \ln x\) because \(4 \times x\) is separated into \(4\) and \(x\).

Step 3: Simplify the expression

The value of \( \ln 4 \) is a constant value ( to be precise it is around 1.386). It is generally left as \( \ln 4 \) unless further simplification is needed. Hence the final expanded expression remains as \(\ln 4 + \ln x\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for expanding logarithmic expressions, solving logarithmic equations, and working with natural logarithms. Logarithms, which are the inverse operations of exponentiation, obey certain rules that make it easier to manipulate mathematical expressions involving them.

Among the most commonly used properties are the product rule, the quotient rule, and the power rule. These properties allow us to expand a single logarithmic expression into a sum or difference of multiple logarithms, as seen with the exercise \(\ln(4x)\), which is expanded using the product rule \(\ln(ab) = \ln(a) + \ln(b)\).

The product rule essentially states that the logarithm of a product is the sum of the logarithms of its factors. Similarly, the quotient rule dictates that the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator. Lastly, the power rule says that the logarithm of a number raised to a power is that power times the logarithm of the number.

• Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• Power Rule: \(\ln(a^b) = b\cdot \ln(a)\)

These properties are pivotal in simplifying logarithmic expressions, which can then be used to further solve equations or interpret the behavior of logarithmic functions.

Solving Logarithmic Equations

Logarithmic equations involve variables within the log function and require leveraging the properties of logarithms for their solutions. They often need a transformation into an exponential form to find the value of the variable. Solving logarithmic equations is a systematic process that may involve expanding, condensing, and using algebraic techniques to isolate the variable.

One common method is to first apply the properties of logarithms to simplify the equation as much as possible. When we have an equation like \(\ln(x) + \ln(y) = \ln(4)\), we could combine the logs on one side by applying the product rule in reverse, as \(\ln(xy) = \ln(4)\), which then suggests that \(xy = 4\) by taking exponentials on both sides.

Another key step could be changing the logarithmic equation into its equivalent exponential form, especially if we have a single log term. For example, \(\ln(x) = 3\) can be rewritten as \(e^3 = x\), where \(e\) is the base of the natural logarithm. With proper understanding and application of logarithmic principles, solving these equations can unravel complex problems in mathematics and applied sciences.

Understanding the Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a specific type of logarithm whose base is Euler's number, \(e\), approximately equal to 2.718281828459. The natural logarithm of a number can be thought of as the power to which \(e\) must be raised to obtain that number.

In other words, if \(\ln(a) = n\), then it follows that \(e^n = a\). This unique property allows the natural logarithm to describe growth processes, decay rates, and time compounding in mathematical finance and other disciplines.

• \(\ln(e) = 1\) because raising \(e\) to the power of 1 yields \(e\).
• \(\ln(1) = 0\) because any number raised to zero is 1, and \(e^0 = 1\).

Understanding the natural logarithm is crucial for various scientific calculations, particularly in disciplines such as physics, biology, and economics, which frequently involve exponential growth or decay. In calculus, \(\ln(x)\) is the inverse function of \(e^x\), and this relationship plays a vital role in integration and differentiation of exponential functions.