Problem 46 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{a}\left(a^{2} \cdot a^{3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Apply the Product Rule for Exponents

First, notice that the expression inside the logarithm can be simplified using the product rule for exponents. The expression \(a^2 \cdot a^3\) can be combined as \(a^{2+3} = a^5\).

Step 2: Use the Power Rule for Logarithms

Now we apply the power rule for logarithms, which states \(\log_b(M^n) = n \cdot \log_b(M)\). For our simplified expression, this becomes \(\log_a(a^5) = 5 \cdot \log_a(a)\).

Step 3: Evaluate the Logarithmic Expression

Recall that the logarithm of a base with itself is 1. That is, \(\log_a(a) = 1\). Thus, \(5 \cdot \log_a(a) = 5 \cdot 1 = 5\).

Key Concepts

Product Rule for Exponents

When we multiply two expressions that have the same base, we can apply the product rule for exponents to simplify them. This rule states that we simply add the exponents of the two expressions. For example, in the expression \(a^2 \cdot a^3\), both parts have the base \(a\). By adding the exponents 2 and 3 together, we get \(a^{2+3} = a^5\). This greatly simplifies our expression and prepares it for further manipulation. Remember that simplifying exponents is an essential step in making logarithmic expressions more manageable.

Power Rule for Logarithms

Once we simplify an expression using exponents, we can then move on to logarithms. The power rule for logarithms allows us to take an exponent and move it in front of the logarithm, thus transforming a complex expression into something simpler. The power rule is expressed as \(\log_b(M^n) = n \cdot \log_b(M)\). In our case with \(\log_a(a^5)\), the exponent 5 goes in front to become \(5 \cdot \log_a(a)\). This turns the logarithmic expression into a more straightforward numerical multiplication, easing the path to finding a solution.

Evaluating Logarithmic Expressions

Finally, to evaluate a logarithmic expression, it's essential to simplify it as much as possible. Often, this involves recognizing that certain logarithmic expressions equate to simple numbers. An important property of logarithms is that the logarithm of a number at its own base equals 1. Thus, for \(\log_a(a)\), the value is 1. This means \(5 \cdot \log_a(a)\) simplifies directly to \(5 \cdot 1 = 5\). Understanding how to evaluate these expressions efficiently will help you solve problems quickly and accurately.