Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$2 \ln e^{4}$$.

Step-by-Step Solution

Answer

The exact value of the logarithm \(2 \ln e^{4}\) is 8.

Step 1: Use the exponent rule of logarithms

The exponent rule of logarithms states that \(\ln m^{n} = n \ln m\). Apply this rule to \( \ln e^{4}\) to get \(4 \ln e\). Therefore, the expression \(2 \ln e^{4}\) can be rewritten as \(2 * (4 \ln e) = 8 \ln e\).

Step 2: Evaluate \(\ln e\)

The natural logarithm of e (\(\ln e\)) equals 1. Thus, replace \(\ln e\) with \(1\) in the previous step.

Step 3: Multiplication

Now that \(\ln e\) has been replaced with 1, we just have to multiply 8 and 1.

Step 4: Final answer

The multiplication of 8 and 1 gives 8.

Key Concepts

Logarithm Properties

Understanding logarithm properties is essential to simplifying logarithmic expressions and solving logarithmic equations. These properties are developed based on the definition of logarithms as inverses of exponentiation. Here are some fundamental properties of logarithms that make problems much easier to handle:

• The Product Rule: Logarithms can turn multiplication into addition, such that \( \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n) \).
• The Quotient Rule: Logarithms can turn division into subtraction, so that \( \text{log}_b \frac{m}{n} = \text{log}_b(m) - \text{log}_b(n) \).
• The Power Rule: Logarithms can turn exponents into multiplication, expressed as \( \text{log}_b(m^n) = n \text{log}_b(m) \). This property is particularly helpful when dealing with expressions like \( \text{ln}(e^4) \).
• Change of Base Formula: It allows computing logarithms with respect to different bases, stated as \( \text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)} \) where c is any positive value.
• The base e Property: For the natural log (ln), if the input is the base e itself, the value is always 1, because \( \text{ln}(e) = 1 \).

These properties work together to allow manipulation of logarithmic statements into simpler forms that are more easily calculated or understood, often making it possible to solve logarithmic problems without a calculator.

Exponent Rule of Logarithms

The exponent rule, also known as the power property of logarithms, is a powerful tool. It simply states that when a logarithm has an exponent, that exponent can be 'brought down' to the front of the log, turning an exponentiation operation within the log into a multiplication operation outside of it. The mathematical expression for this rule is \( \text{log}_b(m^n) = n \text{log}_b(m) \).

In practice, this can vastly simplify calculations. For example, in the given exercise \(2 \text{ln} e^4\), we applied this rule to pull down the exponent, converting it to \(8 \text{ln} e\), which shows how applying this rule makes the problem more straightforward. It's an invaluable trick for solving logarithmic equations, especially when you don't have access to a calculator or when you're working with a base, like e, where you already know the log value.

Logarithms Without a Calculator

Calculating logarithms without a calculator might seem daunting at first, but by using logarithmic properties and rules, one can often find exact values or simplify expressions significantly. For natural logarithms, understanding that the number e and its integral exponents have straightforward natural log values can be a lifesaver. The natural logarithm of e to any power can be easily computed; by default \( \text{ln}(e) = 1 \), and for any integer k, \( \text{ln}(e^k) = k \text{ln}(e) = k \) because of the exponent rule of logarithms.

Similarly, recognizing patterns and relationships, such as logarithms of common fractions or roots, can help in situations where you may not have a calculator. In educational settings, logarithm tables or slide rules were traditionally used before calculators became widespread, although these methods are less precise for complex calculations.

Solving Logarithmic Equations

Solving logarithmic equations can sometimes be straightforward, but when the equations become more complex, knowing how to manipulate and apply the properties and rules of logarithms becomes critical. To solve an equation involving logarithms, first you would isolate the logarithmic part of the equation. You can then use the properties of logarithms to simplify the equation. If the equation has different logarithms on either side but with the same base, you can drop the logarithm and set the inside of the log functions equal to each other.

For instance, if you have \(\text{ln}(x) = \text{ln}(8)\), it simplifies to \(x = 8\). Also, if you can express both sides of the equation as a log of the same base, you can apply the property \(\text{log}_b(m) = \text{log}_b(n)\) if and only if \(m = n\). Additionally, for equations that involve exponents, such as in the exercise provided \(2 \text{ln} e^4\), using the exponent rule can simplify the logarithmic part, which can then be solved as a normal algebraic equation.