Problem 24 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. \(\ln 0.03\)

Step-by-Step Solution

Answer

\(\ln 0.03 \approx -3.5066\)

Step 1: Understanding Natural Logarithms

The expression "\( \ln 0.03 \)" involves the natural logarithm function. Natural logarithms are logarithms to the base \( e \), where \( e \approx 2.71828 \). The natural logarithm of a number gives a power to which \( e \) must be raised to get that number.

Step 2: Using a Calculator

To find \( \ln 0.03 \), use a calculator with natural logarithm functionality. Enter "0.03" into the calculator and press the "ln" button.

Step 3: Rounding the Result

The calculator should display the result of \( \ln 0.03 \). Round this result to four decimal places to get the final answer.

Key Concepts

Logarithms

Logarithms are mathematical functions that help us determine the power needed to raise a base number to achieve a certain value. They are the inverse operations of exponentiation. In other words, if you have an equation of the form \( b^x = y \), then the logarithm tells us \( x \), which is the power to which \( b \), the base, must be raised to obtain \( y \).

• Common Bases: Logarithms usually have a base of 10 (common logarithms), \( e \) (natural logarithms), or 2 (binary logarithms).
• Natural Logarithms: These have the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828. The notation \( \ln(y) \) represents the natural logarithm of \( y \).

Using natural logarithms can simplify expressions involving exponentials, especially those related to growth and decay in calculus, biology, and even finance. To compute natural logarithms without a calculator, you often need to rely on approximations or look-up tables, especially when the result is not an integer.
Natural logarithms are widely used due to their unique properties when dealing with rates of change.

Exponential Functions

Exponential functions are a type of function where the variable is in the exponent. The general form of an exponential function is \( f(x) = a \cdot b^x \), where \( a \) is a constant, and \( b \) is the base.

• Key Characteristic: Exponential growth happens when the base \( b \) is greater than 1, leading to rapid increases.
• Exponential Decay: Occurs when \( 0 < b < 1 \), leading to a decrease towards zero.

The natural exponential function is \( e^x \). The constant \( e \) plays a significant role due to its natural occurrence in continuous growth processes. Inversely, natural logarithms transform exponential equations into linear ones, simplifying calculations and interpretations.
The equation \( \ln(y) = x \) indicates that \( y = e^x \), and this connection is fundamental in many scientific and engineering applications to simplify shapes of curves or solve equations related to exponentials. It allows us to model phenomena where growth or decay rates are proportional to the current value, such as in population dynamics or radioactive decay.

Rounding Numbers

Rounding numbers is a mathematical process to reduce the number of digits in a number while keeping it close to the original value. It's crucial in approximations to make numbers easier to handle and interpret without significantly losing accuracy.

• Purpose: It helps simplify numbers, especially in cases where precision to many decimal points is unnecessary.
• Methods: Round up, round down, or round to the nearest, depending on the context and specified rules.

In the context of the expression \( \ln 0.03 \), rounding is used to round the calculated value to four decimal places for precision and simplicity. This kind of rounding maintains a balance between accuracy and readability, making it easy to compare results or use them in further calculations.
When rounding to four decimal places, you look at the fifth decimal place: if the digit is 5 or more, you round up; if it is less than 5, you round down. This simple rule ensures consistency across calculations and reporting of results.