Problem 45 - Algebra 2 and Trigonometry

Question

In \(45-52,\) if \(\ln a=c,\) express each of the following in terms of \(c\) $$ \ln a^{2} $$

Step-by-Step Solution

Answer

The expression \( \ln a^2 \) in terms of \( c \) is \( 2c \).

Step 1: Understanding the expression

The problem is asking to express \( \ln a^2 \) in terms of \( c \), given that \( \ln a = c \). We need to utilize logarithmic properties to rewrite the expression.

Step 2: Applying the Power Rule of Logarithms

The power rule of logarithms states that \( \ln a^b = b \cdot \ln a \). Here, we have \( a^2 \), so the expression can be rewritten as \( 2 \cdot \ln a \).

Step 3: Substituting for \( \ln a \)

We know from the problem statement that \( \ln a = c \). Substitute \( c \) for \( \ln a \) in the expression from Step 2. Thus, \( 2 \cdot \ln a = 2c \).

Step 4: Final Expression

Now that we've made the substitution, the expression in terms of \( c \) is \( 2c \).

Key Concepts

Properties of Logarithms

Logarithms can be tricky at first, but they are actually very logical once you get the hang of them. A logarithm answers the question: "To what exponent must the base be raised, to produce a certain number?" There are several key properties that make them easier to work with.

First, let's talk about the **product property**:

• This states that the logarithm of a product is the sum of the logarithms: \[\ln(ab) = \ln(a) + \ln(b)\]

This helps simplify logs of products.

Next, there's the **quotient property**:

• This states that the logarithm of a quotient is the difference of the logarithms:\[\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\]

Finally, the property most relevant to our original problem is the **power rule**. These properties are super handy for transforming logarithmic expressions into simpler forms.

Power Rule of Logarithms

This is one of the most essential properties of logarithms. It allows you to move the exponent in a powered term to the front, making calculations much easier. For any positive number \(a\) and any real number \(b\), the power rule is written as:

• \[\ln(a^b) = b \cdot \ln(a)\]

In the problem we had \( \ln a^2 \). Using the power rule, we take the exponent \(2\) and shift it to multiply in front of the logarithm term. This transforms the expression into \( 2 \cdot \ln a \).
A powerful tool for simplifying! It simplifies logs and aids calculation.

Expressions in Terms of Variables

When working with expressions, sometimes you're asked to rewrite them in terms of another variable. This often uses substitution based on given information. In the example, you were told \( \ln a = c \). This is a piece of given data that lets us transform the expression using substitution.

Think of it like a puzzle. You've identified that wherever you see \( \ln a \), you can plug in \(c\). In our problem, after applying the power rule to obtain \( 2 \cdot \ln a \), we simply replaced \( \ln a \) with \( c \), resulting in the expression \( 2c \). This type of variable substitution is a powerful technique in simplifying expressions and finding solutions.

Exponential Functions

Exponential functions are fundamental in mathematics. They express relationships where a constant base is raised to a variable exponent, producing steep changes. The general form is:

• \[y = a^x\]

where \(a\) is a constant, and \(x\) is the exponent variable. They grow rapidly and are inversed by logarithms.

In our problem, \( a^2 \) is part of the exponential function setup. Understanding that \( a \) is raised to a power simplifies work with logs. Calculating with exponentials and logs reveals patterns in growth and decay, helping solve many real-world problems.