Problem 22 - Algebra and Trigonometry

Question

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } e^{\ln \pi}} & {\text { (b) } 10^{\log 5}} & {\text { (c) } 10^{\log 87}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\pi\); (b) 5; (c) 87.

Step 1: Understand the Expression in (a)

The expression in part (a) is \( e^{\ln \pi} \). Recall that the natural logarithm function \( \ln \) and the exponential function \( e^x \) are inverse functions. This means that \( e^{\ln x} = x \) for any positive value \( x \).

Step 2: Apply the Inverse Property in (a)

Since \( e^{\ln x} = x \), we can apply it to \( e^{\ln \pi} \) to conclude that \( e^{\ln \pi} = \pi \).

Step 3: Understand the Expression in (b)

The expression in part (b) is \( 10^{\log 5} \). Similarly, the logarithmic function \( \log \) (base 10) and the exponential function \( 10^x \) are inverse functions. Thus, \( 10^{\log x} = x \) for \( x > 0 \).

Step 4: Apply the Inverse Property in (b)

By applying the property \( 10^{\log x} = x \), we find \( 10^{\log 5} = 5 \).

Step 5: Understand the Expression in (c)

For part (c), we have the expression \( 10^{\log 87} \). Again, use the property that \( 10^{\log x} = x \).

Step 6: Apply the Inverse Property in (c)

Using the inverse property here gives us \( 10^{\log 87} = 87 \).

Key Concepts

Exponential Function

The exponential function is a powerful mathematical tool denoted typically as \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. This function arises frequently in various scientific fields including finance, physics, and biology because it models growth and decay processes efficiently.

Some key properties of exponential functions include:

• They are always positive, meaning \( e^x > 0 \) for all real numbers \( x \).
• The function grows faster as \( x \) increases, meaning it's non-linear and not restricted to a specific rate of increase.
• It is the inverse of the natural logarithm function \( \ln x \).

Understanding the exponential function helps in solving expressions like \( e^{\ln x} \). Here, knowing that \( e \) and \( \ln x \) are inverse functions will simplify evaluations since \( e^{\ln x} = x \).
In practice, this means expressions such as \( e^{\ln \pi} \) simplify directly to \( \pi \), due to the inverse relationships.

Natural Logarithm

The natural logarithm \( \ln x \) is another fundamental tool in mathematics, defined as the logarithm with base \( e \). This function helps transform complex multiplications into simpler additions and is the inverse of the exponential function.

Some essential properties include:

• \( \ln 1 = 0 \) because \( e^0 = 1 \).
• \( \ln(e^x) = x \) helping to solve equations where the exponent is unknown.
• The value of \( \ln x \) is only defined for positive \( x \), echoing the positivity constraint of exponential functions.

One practical implication of understanding the natural logarithm is its use in solving expressions like \( e^{\ln x} = x \). Knowing \( \ln \) is the inverse to exponential allows these transformations to happen intuitively, simplifying calculations such as \( e^{\ln \pi} = \pi \). Hence, understanding logarithms is key to mastering many mathematical problems.

Logarithmic Function

A logarithmic function is crucial for solving equations that involve exponential expressions. The common logarithm \( \log x \) usually refers to a base 10 log, indicated as \( \log_{10} x \). It is the inverse of the exponential function \( 10^x \).

Significant properties of logarithmic functions include:

• \( \log_{10} 1 = 0 \) because \( 10^0 = 1 \).
• \( \log_{10} (10^x) = x \), showing the inverse relationship mentioned earlier.
• These functions are undefined for non-positive values, aligning with the nature of their respective exponential counterparts.

Utilizing the properties of logarithmic functions simplifies calculations, such as solving \( 10^{\log x} = x \). When applied to specific expressions like \( 10^{\log 5} \), the inverse function property instantly reveals the solution \( x = 5 \). Understanding these principles makes it easier to manage and evaluate logarithmic expressions and their exponents reliably.