Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \ln 1,000^{2} $$

Step-by-Step Solution

Answer

The evaluated logarithm is 13.82.

Step 1: Understand the Property of Logarithms

Recall the logarithmic property: \( \ln(a^b) = b \cdot \ln(a) \). This allows us to handle exponents inside a logarithm.

Step 2: Apply the Property to the Given Expression

The expression is \( \ln(1000^2) \). Applying the property gives us: \( 2 \cdot \ln(1000) \).

Step 3: Calculate the Natural Logarithm of 1000

Use a calculator to find \( \ln(1000) \). The natural logarithm of 1000 is approximately 6.908.

Step 4: Multiply to Find the Final Result

Multiply the result from Step 3 by 2: \( 2 \cdot 6.908 = 13.816 \).

Step 5: Round the Result to the Nearest Hundredth

Round 13.816 to the nearest hundredth. The nearest hundredth is 13.82.

Key Concepts

Logarithmic Properties

Logarithmic properties are fundamental rules that arise from the nature of logarithms. They help simplify expressions and solve logarithmic equations easily. One essential property is named the "Power Rule," which states that the logarithm of a power can be simplified by multiplying the exponent by the logarithm of the base. If you have a logarithm such as \( \ln(a^b) \), this property tells us it equals \( b \cdot \ln(a) \). This fact is particularly useful when you need to evaluate expressions with exponents inside a logarithm, just like in our exercise where we handle \( \ln(1000^2) \) and simplify it to \( 2 \cdot \ln(1000) \). Understanding these properties is key to efficiently working with logarithms.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special kind of logarithm with the base \( e \), where \( e \) is approximately equal to 2.718. It is widely used in various mathematical equations and applications, especially those involving growth or decay, like compound interest or population models. The natural logarithm has unique properties, such as:

• \( \ln(1)= 0 \)
• \( \ln(e) = 1 \)

To solve for \( \ln(1000) \), you can use a calculator, which gives an approximate value of 6.908. This number is key to further calculations in the step-by-step process. By understanding the natural logarithm and using it effectively, collapsing complex expressions becomes simpler.

Exponents

Exponents are numbers written as a superscript to signify repeated multiplication of a base number. For example, \(1000^2\) represents multiplying 1000 by itself: \(1000 \times 1000\). Exponents are integral when dealing with logarithms because one of the main logarithmic properties involves exponents: the previously mentioned Power Rule. In logarithmic problems, understanding how to handle exponents can streamline the process of finding solutions.

• Exponential notation helps in expressing large numbers succinctly.
• They link closely with logarithms since a logarithm is essentially an exponent in disguise.

When dealing with expressions like \( \ln(1000^2) \), recognizing the exponent allows you to directly apply logarithmic properties for simplification, thereby making calculations much more manageable.