Problem 43 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\sqrt{7} \ln e^{\sqrt{7}}$$

Step-by-Step Solution

Answer

The expression evaluates to 7.

Step 1: Simplify the Exponent

The given expression is \( \sqrt{7} \ln e^{\sqrt{7}} \). Note that the notation \( \ln e^x \) can be simplified using the property of logarithms: \( \ln e^x = x \). Thus, the expression \( \ln e^{\sqrt{7}} \) simplifies to \( \sqrt{7} \).

Step 2: Multiply the Simplified Terms

Substitute \( \ln e^{\sqrt{7}} \) with \( \sqrt{7} \) in the original expression. This yields a new expression: \( \sqrt{7} \times \sqrt{7} \).

Step 3: Evaluate the Multiplication of Square Roots

To evaluate \( \sqrt{7} \times \sqrt{7} \), use the property \( (\sqrt{a})^2 = a \). Therefore, \( \sqrt{7} \times \sqrt{7} = 7 \).

Key Concepts

Properties of Logarithms

Logarithms are a mathematical tool used to simplify calculations involving exponentials. One of their most useful properties is the ability to simplify expressions like \( \ln e^x = x \). This property arises because the logarithm with base \( e \), known as the natural logarithm, is the inverse operation of raising \( e \) to a power.

In simpler terms:

• If you take \( e \) and raise it to a power \( x \), then take the natural logarithm of the result, you'll end up with \( x \) again. This is why \( \ln e^x \) simplifies to \( x \).

The properties of logarithms, like this one, are vital for simplifying complex expressions and solving equations that would otherwise be cumbersome to handle without a calculator.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that makes complex problems more manageable. The goal is to rewrite expressions in the simplest form possible so that solving or evaluating them becomes easier.

Using the properties of logarithms, we simplified the expression \( \ln e^{\sqrt{7}} \) to just \( \sqrt{7} \). This allowed us to handle the expression \( \sqrt{7} \ln e^{\sqrt{7}} \) by turning it into \( \sqrt{7} \times \sqrt{7} \).

In general, always look for opportunities to:

• Combine like terms by addition or subtraction.
• Use properties of exponents and logarithms to simplify exponential or logarithmic expressions.
• Factor expressions when possible to make multiplication and division straightforward.

This makes the overall evaluation process simpler and more efficient.

Exponents

Exponents are used to denote repeated multiplication. For instance, \( e^x \) means \( e \) multiplied by itself \( x \) times. Understanding properties of exponents is crucial for simplifying expressions involving powers and logarithms.

One important property we've seen is how an exponential and its corresponding logarithm cancel each other out, as shown in \( \ln e^x = x \). This cancellation is instrumental in simplifying expressions and equations involving exponential growth and decay.

Key points to remember about exponents:

• \( (a^m)^n = a^{m \times n} \)
• \( a^m \times a^n = a^{m+n} \)
• \( a^0 = 1 \) for any non-zero \( a \)

These properties enable you to manipulate expressions containing exponents, simplifying your work significantly.

Square Roots

Square roots are a special kind of radical notation used to represent a number that, when multiplied by itself, gives the original number. If a square has an area of \( x \), the length of one side is the square root of \( x \), denoted as \( \sqrt{x} \).

In our solution, we encountered \( \sqrt{7} \), and eventually \( \sqrt{7} \times \sqrt{7} \). A fundamental property of square roots is \( (\sqrt{a})^2 = a \), meaning a square root multiplied by itself yields the original number.

Here are some quick tips:

• Square roots can only be taken of non-negative numbers in real numbers.
• \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)
• \( \sqrt{a/b} = \sqrt{a}/\sqrt{b} \)

These properties are often handy when simplifying expressions, ensuring calculations are accurate and clean.