Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \ln 5+\ln 7 $$

Step-by-Step Solution

Answer

The evaluated expression is approximately 3.56.

Step 1: Apply Logarithm Product Rule

The expression given is \( \ln 5 + \ln 7 \). The product rule for logarithms states that \( \ln a + \ln b = \ln (a \cdot b) \). Therefore, apply the product rule: \( \ln 5 + \ln 7 = \ln (5 \cdot 7) \).

Step 2: Compute the Product

Calculate the product inside the logarithm: \( 5 \times 7 = 35 \). Thus, the expression simplifies to \( \ln 35 \).

Step 3: Evaluate the Natural Logarithm

Use a calculator to evaluate \( \ln 35 \) to the nearest hundredth. Upon calculation, \( \ln 35 \approx 3.56 \).

Key Concepts

Logarithm Product Rule

Logarithms help us solve problems involving multiplication and division more easily. They do this by transforming those problems into addition and subtraction. One of the key tools in this transformation is the logarithm product rule.
The logarithm product rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Formally, this is expressed as

• \( \ln a + \ln b = \ln (a \cdot b) \)

This property is essential because it simplifies complex expressions, making calculations easier.
For example, if you have \( \ a = 5 \) and \( b = 7\), using this rule, you find

• \( \ \ \ \ \ \ \ \ \ln 5 + \ln 7 = \) \( \ln (5 \cdot 7) = \) \( \ln 35 \)

By applying the product rule, it's simpler to work with one single logarithm of \(35\) than two separate logarithms.

Natural Logarithm

Not all logarithms are created equal. The natural logarithm is special and quite common in mathematics, especially in calculus and exponential growth problems. A natural logarithm is a logarithm with base \(e\), where \(e\) is approximately \(2.71828\).
When evaluating expressions like \( \ln 5 \), you're working with natural logarithms, which are often abbreviated as \"ln\".

• These logarithms are tuned to the natural base \(e\) and are particularly useful for continuous growth processes, such as exponential growth or radioactive decay.
• The natural logarithm of any number represents the time needed to grow to that number with continuous compounding, starting from \(1\).

This makes natural logarithms very intuitive and valuable in fields like biology, economics, and engineering, where growth is often continuous.

Evaluating Logarithms

Once you have a simplified expression using the product rule, like \( \ln 35 \), it's time to evaluate it.
This means finding its numerical value, usually with a calculator, especially when it's not a simple number. Calculators have built-in functions for natural logarithms, which makes evaluation easy.

• Enter the expression \( \ln 35 \) to get its approximate value.
• For this particular exercise, rounding it to the nearest hundredth gives you \( \ln 35 \approx 3.56 \).

Evaluating logarithms accurately is crucial in achieving precise results, especially when you rely on them to solve real-world problems or continue further calculations. This is why using a calculator for more complex numbers is a standard practice in mathematical analysis and computations.