Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Simplify each expression. (a) \(\log _{3} 3^{19}\) (b) \(\log _{4} 4^{17}\) (c) \(\log _{12} 12^{1 / 3}\) (d) \(\log _{a} \sqrt{a}(a>0, a \neq 1)\)

Step-by-Step Solution

Answer

(a) 19, (b) 17, (c) 1/3, (d) 1/2

Step 1: Understand the Logarithm Power Rule

The power rule of logarithms states that \( \log_{b}(a^c) = c\log_{b}(a) \). This rule is very useful in simplifying expressions where the argument is an exponent.

Step 2: Simplify expression (a)

Using the power rule, we have \( \log_{3} 3^{19} = 19\log_{3}3 \). Since \( \log_{3}3 = 1 \), the expression simplifies to 19.

Step 3: Simplify expression (b)

Apply the power rule: \( \log_{4} 4^{17} = 17\log_{4}4 \). Knowing that \( \log_{4}4 = 1 \), the expression simplifies to 17.

Step 4: Simplify expression (c)

Apply the power rule: \( \log_{12} 12^{1/3} = \frac{1}{3}\log_{12}12 \). Since \( \log_{12}12 = 1 \), the expression simplifies to \( \frac{1}{3} \).

Step 5: Simplify expression (d)

Rewrite the square root as a power: \( \sqrt{a} = a^{1/2} \). Then, apply the power rule: \( \log_{a} a^{1/2} = \frac{1}{2}\log_{a}a \). As \( \log_{a}a = 1 \), the expression simplifies to \( \frac{1}{2} \).

Key Concepts

Exponents

Exponents can be thought of as shorthand for repeated multiplication of the same number by itself. When we use exponents, it means multiplying the base number several times. For instance, using our number 3, if you see it raised to the 2nd power, written as \(3^2\), it means you're multiplying 3 by itself: \(3 \times 3\). Similarly, \(3^3\) means \(3 \times 3 \times 3\).
In logarithms, exponents have a critical role. The logarithm of a number is asking "to what exponent (or power) must we raise the base to get this number?" For example, if you ask for \(\log_3(9)\), you are essentially being asked "3 raised to what power equals 9?" The answer here is 2 because \(3^2 = 9\).
Understanding exponents is foundational in dealing with powers of numbers and thus crucial for easily managing logarithmic problems.

Simplification

Simplification of expressions involves reducing them to their simplest form, making them easier to work with or understand. When working with logarithmic expressions, simplification often involves employing rules of logarithms to condense and clarify expressions.
With expressions like \(\log_b(b^c)\), simplification becomes direct once we recognize patterns. The power rule plays a major part here, allowing us to take what might seem mathematically burdensome expressions and simplify them.

• Convert complex roots or powers into simpler forms whenever possible. For instance, rewriting \(\sqrt{a}\) as \(a^{1/2}\) helps in applying the power rule effortlessly.
• Always seek to identify if the logarithmic and exponential bases are the same, as this knowledge instantly simplifies the expression.

Using these strategies, you can transform potentially daunting expressions into straightforward solutions.

Power Rule

The power rule of logarithms is a powerful tool used to simplify expressions where the argument is an exponent. This rule states that if you have a logarithm expression of the form \(\log_{b}(a^c)\), it equals \(c \log_{b}(a)\). In simpler terms, you can "bring down" the exponent as a multiplier in front of the log.
Let's look at how this applies:
In the expression \(\log_{4} 4^{17}\), using the power rule, we move 17 in front of the logarithm, which simplifies it to \(17 \log_{4} 4\). Since \(\log_{4} 4\) is 1, the expression further simplifies to 17. This rule is valuable as it converts tricky exponential arguments into simpler arithmetic problems.

• The power rule shines when the base of the log and the base of the exponent match, instantly reducing complexity.
• Remember that this rule doesn't change the mathematical balance; it simply offers an easier path to the answer.

Mastering the power rule means you'll find logarithmic simplifications much more manageable and intuitive.