Problem 24 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{6} \sqrt[3]{6}$$

Step-by-Step Solution

Answer

The exact value of \(\log_{6}\sqrt[3]{6}\) is \(1/3\).

Step 1: Understanding Cube Roots

Express the cubic root of 6 as a power of 6. Recall that the cube root of a number \(n\) is equivalent to raising \(n\) to the power of \(1/3\). Therefore, we can write \(\sqrt[3]{6}\) as \(6^{1/3}\).

Step 2: Solve using Logarithm Properties

The logarithm property \(\log_{b}(a^m) = m \log_{b}a\) states that the exponent on the argument of a logarithm can be brought out front and multiplied by the logarithm. Applying this property to our expression, we have \(\log_{6}6^{1/3} = (1/3) \log_{6}6\).

Step 3: Compute the Logarithm

Finally, compute the logarithm \(\log_{6}6\). Any logarithm base \(b\) of \(b\) is always 1, so \(\log_{6}6 = 1\), since \(6^1 = 6\). Multiplying this by \(1/3\), we get the result.

Key Concepts

Properties of Logarithms

Logarithms have several useful properties that make calculations simpler. Understanding these properties lets us transform complex logarithmic expressions into more manageable forms. Here are some key properties you should know:

• Power Rule: The property \(\log_{b}(a^m) = m \log_{b} a\) tells us how to manage expressions that contain exponents within a logarithm. This allows you to bring the exponent, \(m\), in front of the logarithmic expression as a product.
• Product Rule: The property \(\log_{b}(mn) = \log_{b} m + \log_{b} n\) helps in splitting the logarithm of a product into the sum of two logarithms.
• Quotient Rule: The property \(\log_{b}\left(\frac{m}{n}\right) = \log_{b} m - \log_{b} n\) allows for the division inside a logarithm to be transformed into the difference of two logarithms.

These properties are all based on the rules of exponents because logarithms are essentially the inverse operations of exponentiation. Using the power rule was crucial in finding the value of the expression \(\log_{6} \sqrt[3]{6}\) in our exercise.

Cube Roots

Cube roots are a specific type of root used frequently in mathematics. When you take the cube root of a number, you are looking for a value that, when multiplied by itself three times, gives the original number.To express a cube root as an exponent, we use powers. For example, the cube root of 6 is expressed as \(\sqrt[3]{6}\), which is equivalent to \(6^{1/3}\). This fractional exponent representation is helpful in logarithmic calculations because it allows us to apply logarithm properties, like the power rule, to simplify expressions.Understanding how roots and exponents relate is important because it opens up the ability to move between these two forms easily. This is evident in our task where \(\sqrt[3]{6}\) was transformed to \(6^{1/3}\) to utilize logarithmic properties effectively.

Exponents

Exponents are a foundational part of algebra that describe how many times a number, known as the base, is multiplied by itself. For example, in the term \(b^n\), \(b\) is the base and \(n\) is the exponent. Exponents can also be fractional, where the numerator indicates a power and the denominator indicates a root. A common example is the expression \(b^{1/3}\), which corresponds to the cube root of \(b\). This concept links closely with how we computed \(\log_{6} \sqrt[3]{6}\) by rewriting the cubic root as an exponent.Knowing how to manipulate expressions with exponents (including fractional ones) is vital for using logarithms effectively. It helps when applying properties like the power rule, simplifying the calculation process and solving expressions without a calculator.