Problem 21 - Precalculus

Question

Evaluate the expression. \(\log _{36}(216)\)

Step-by-Step Solution

Answer

\( \log_{36}(216) = \frac{3}{2} \).

Step 1: Convert to a Common Base

To solve \( \log_{36}(216) \), we start by recognizing both 36 and 216 in terms of powers of 6: \( 36 = 6^2 \) and \( 216 = 6^3 \).Let's rewrite the logarithmic expression using the common base 6:\( \log_{36}(216) = \log_{6^2}(6^3) \).

Step 2: Apply the Change of Base Formula

By applying the power rule for logarithms, we can simplify further.\[ \log_{a^m}(a^n) = \frac{n}{m} \log_a(a) \].Since \( \log_a(a) = 1 \):\( \log_{6^2}(6^3) = \frac{3}{2} \log_6(6) = \frac{3}{2} \).

Step 3: Final Simplification

Since the expression simplifies directly to a numerical value, this step confirms that the evaluation of the original logarithmic expression is complete.Thus, \( \log_{36}(216) = \frac{3}{2} \).

Key Concepts

Change of Base Formula

Understanding logarithms can be a bit tricky, but thankfully, the change of base formula makes working with them more manageable. This handy formula allows us to convert a logarithm to different bases, which often simplifies calculations. The formula is given by:

• \( \log_{b}(a) = \frac{ \log_{c}(a) }{ \log_{c}(b) } \)

Here, \( b \) and \( a \) are the given base and number, and \( c \) is the new base you choose to convert to, usually 10 or \( e \) (Euler's number), depending on whether you're using common or natural logarithms. This conversion helps because it allows us to use calculators or computational tools that easily handle base 10 or base \( e \) logarithms. It's essential for simplifying complex logs and making them easier to compute by hand when needed.

In our example, instead of using an entirely different base, we simplified using a similar concept focused on the powers of numbers, turning the expression into one involving the same base, making it straightforward to solve.

Powers of Numbers

When dealing with logarithmic expressions, understanding powers of numbers is crucial. Powers, or exponents, are a way to express repeated multiplication of a number by itself. For instance, \( 6^2 = 36 \) as 6 multiplied by itself equals 36. Similarly, \( 6^3 = 216 \) because 6 multiplied by itself three times yields 216.

This knowledge is immensely helpful in converting numbers into a common base, as seen in our exercise. By recognizing both 36 and 216 as powers of 6, we can express the original logarithmic problem in terms of a simpler base:

• \( 36 = 6^2 \)
• \( 216 = 6^3 \)

Converting to a common base helps simplify the logarithmic calculation process. It makes use of the logarithm properties, like the power rule, which we'll delve into next. By simplifying expressions using powers, we make complex problems more accessible and easier to solve.

Logarithm Properties

Logarithm properties are essential tools when working with logarithmic expressions. They help us manipulate and simplify these expressions for easier evaluation. One critical property is the power rule for logarithms, which states:

• \( \log_{b}(a^n) = n \cdot \log_{b}(a) \)

This rule tells us that when the number inside the logarithm is raised to a power, we can bring the power out front as a multiplier. It's this property we see applied in solving our exercise:

• \( \log_{6^2}(6^3) = \frac{3}{2} \cdot \log_6(6) \)

Since \( \log_6(6) = 1 \), the expression further simplifies to \( \frac{3}{2} \). Understanding and using these properties makes handling logarithms much easier.

Besides the power rule, other key properties like the product, quotient, and change of base formulas are vital when dealing with logarithms. Recognizing these allows us to break down complex logarithmic expressions into simpler, more manageable parts. By doing so, we can solve problems efficiently and accurately.