Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln e^1.6

Verified step by step guidance

Recognize that the problem involves the natural logarithm function, denoted as \( \ln \), and the exponential function with base \( e \).

Recall the property of logarithms: \( \ln(e^x) = x \). This property is derived from the fact that the natural logarithm is the inverse of the exponential function with base \( e \).

Apply this property to the given expression: \( \ln(e^{1.6}) \).

According to the property, \( \ln(e^{1.6}) = 1.6 \).

Since the problem asks for an approximation to four decimal places, note that 1.6 is already a decimal number with one decimal place, and no further approximation is needed.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Natural Logarithm (ln)

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is an irrational constant approximately equal to 2.71828. It is used to solve equations involving exponential growth or decay. The natural logarithm has the property that ln(e^x) = x, which simplifies calculations involving exponential functions.

Exponential Function

An exponential function is a mathematical function of the form f(x) = a * e^(bx), where 'a' and 'b' are constants, and 'e' is the base of the natural logarithm. These functions model growth or decay processes, such as population growth or radioactive decay. Understanding how to manipulate and evaluate exponential functions is crucial for solving logarithmic equations.

Approximation and Rounding

Approximation involves estimating a value that is close to the actual number, often used when exact values are difficult to compute or unnecessary. Rounding to four decimal places means adjusting a number so that it has four digits after the decimal point, which is important for clarity and precision in mathematical results. This concept is essential when presenting final answers in a concise format.

Watch next

Master Product, Quotient, and Power Rules of Logs with a bite sized video explanation from Patrick

Start learning