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Chapter 5: Problem 26

Find the exact value of the expression without using your GDC. $$\ln e^{13}$$

Short Answer

Expert verified
The exact value of \( \ln e^{13} \) is 13.

Step by step solution

01

Understand the natural logarithm properties

Remember that the natural logarithm function, denoted as \( \ln \), is the inverse of the exponential function \( e^x \). This is important because \( \ln(e^x) = x \). This property will be used to solve the expression \( \ln e^{13} \).
02

Apply the property to simplify

Using the property \( \ln(e^x) = x \), substitute \( x = 13 \) into the expression. So \( \ln(e^{13}) = 13 \) since the natural logarithm and the exponential function \( e^x \) are inverses.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Function
In mathematics, an inverse function essentially reverses another function's output back to its input. This is a powerful tool because it lets us "undo" actions of functions, making our work with equations and expressions more flexible and manageable. Let's break this down further:
  • Suppose you have a function \( f(x) \). The inverse function, denoted as \( f^{-1}(x) \), will "reverse" \( f(x) \) such that if \( y = f(x) \), then \( x = f^{-1}(y) \).
  • In essence, applying a function and then its inverse gets you back to where you started: \( f(f^{-1}(x)) = x \).
The natural logarithm and the exponential function are examples of inverse functions. This means that for any number \( x \), the function \( ln(e^x) \) simplifies back to just \( x \). This understanding of inverse functions allows us to handle expressions like \( ln(e^{13}) \) effortlessly by reducing it to \( 13 \).
Exponential Function
The exponential function is one of the most important mathematical functions. It deals with growth, decay and a natural base, denoted by \( e \), which is approximately 2.718. Here's why it's special:
  • The function has the form \( f(x) = e^x \), where \( x \) is any real number.
  • It is known for its unique property where the rate of growth (or decay) at any point is equal to its value at that point. This property makes it extremely useful in real-world scenarios like compound interest, population growth, and radioactive decay.
When solving expressions involving \( e \), such as \( e^{13} \), remember that this expression signifies the exponential growing at a particular power, 13 in this case. The harmony between this function and its inverse, the natural logarithm, allows us to simplify expressions like \( ln(e^{13}) \) effectively.
Logarithmic Properties
Understanding logarithmic properties is essential to working with logarithms efficiently. These properties make it easier to simplify and manipulate logarithmic expressions. Let's consider some key properties:
  • The **Product Rule**: \( ln(ab) = ln(a) + ln(b) \), which allows us to split products into sums.
  • The **Quotient Rule**: \( ln \left( \frac{a}{b} \right) = ln(a) - ln(b) \), enabling division to be converted into subtraction.
  • The **Power Rule**: \( ln(a^b) = b \, ln(a) \), where exponents can be brought down in front as coefficients.
  • The **Change of Base Formula**: Helps us convert between different logarithm bases, such as from base 10 to base \( e \): \( log_b(a) = \frac{ln(a)}{ln(b)} \).
Applying these properties, you can address and simplify many real-world problems involving logarithms, making it an essential tool in a mathematician's toolkit.
In the context of our example \( ln(e^{13}) \), leveraging the inverse rule of the natural logarithm function directly simplifies it to 13 with no further transformations needed.

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Most popular questions from this chapter

Use properties of logarithms to write each logarithmic expression as a sum, difference and/or constant multiple of simple logarithms (i.e. logarithms without sums, products, quotients or exponents). $$\log \left(\frac{9}{x}\right)$$

Solve for \(x\) in the logarithmic equation. Give exact answers and be sure to check for extraneous solutions. $$\log x^{8}=(\log x)^{4}$$

Find the exact value of the expression without using your GDC. $$\log _{3}\left(\frac{1}{27}\right)$$

Sketch a graph of the function and state its domain, range, \(y\) -intercept and the equation of its horizontal asymptote. $$q(x)=3\left(3^{-x}\right)-3$$

Find the exact value of the expression without using your GDC. $$\ln \left(\frac{1}{e^{2}}\right)$$
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