Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of natural logarithms to rewrite the expression. $$7 \ln e^{0}$$

Step-by-Step Solution

Answer

After using properties of logarithms and simplifying the expression, the result is 0.

Step 1: Use property of exponents

Firstly, focus on the term \(e^{0}\). One of the primary rules of exponentiation states that any non-zero number to the power of 0 equals 1. Thus, \(e^{0} = 1\) . Substitute this into the expression.

Step 2: Simplify the natural logarithm

Substitution leads to the new expression \(7 \ln 1\). The natural logarithm of 1, or any logarithm of 1, regardless of base, is always 0 since any number to the power of 0 equals 1. Thus, \( \ln 1 = 0\), substitute this back into the expression.

Step 3: Final simplifications

After replacing \( ln 1 \) with 0 in the expression, we get \(7 * 0\), which equals 0.

Key Concepts

Properties of Exponents

Exponents are mathematical operations that deal with powers. One of the fundamental properties of exponents is when any non-zero number is raised to the power of zero. This property states that for any number \(a\), \(a^0 = 1\). This might seem surprising at first, but it aligns with the pattern of dividing a number by itself as you decrease the exponent by one. For example:

• \(a^3 = a \times a \times a\)
• \(a^2 = a \times a\)
• \(a^1 = a\)

Now, dividing \(a\) by \(a\) results in 1, which is why \(a^0 = 1\). This property is essential because it simplifies expressions, like the one observed in the original exercise, where \(e^0 = 1\). Understanding these properties helps in transforming complex exponential expressions into more manageable forms.

Simplifying Logarithms

Logarithms are the inverse operations of exponentiation. Simplifying them involves using properties to make expressions simpler. One key idea is understanding what the natural logarithm represents. The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828.When simplifying a logarithm of a constant, like \(\ln 1\), remember that the logarithm asks "to what power should the base be raised, to obtain this number?" Since any number to the power of zero is 1, \(\ln 1 = 0\). Knowing this property lets us simplify expressions significantly. Applying this to the expression \(7 \ln 1\) turns it into \(7 \times 0\), which is much simpler. Mastering these simplifications is crucial for handling more complex logarithmic expressions.

Logarithmic Identities

Logarithmic identities are mathematical rules that allow us to manipulate and simplify logarithmic expressions. These identities are very useful for solving logarithmic equations and understanding the behavior of logarithmic functions. A fundamental identity is \(\log_b(1) = 0\) for any base \(b\), which translates to \(\ln(1) = 0\) when the base is \(e\).Another vital identity associated with natural logarithms is \(\ln(e) = 1\). This is because \(e\) raised to the first power is itself. These identities help make sense of logarithmic expressions and are instrumental in the step-by-step solutions we apply, as seen in calculating \(7 \ln 1\). Using identities, such expressions collapse into much simpler forms, facilitating easier computation and deeper understanding.