Problem 7 - College Algebra

Question

Write each equation in its equivalent exponential form. $$ \log _{6} 216=y $$

Step-by-Step Solution

Answer

The equivalent exponential form of \(\log _{6} 216 = y\) is \(6^y = 216\).

Step 1: Identify the base, exponent and result

In the logarithm \(\log _{6} 216=y\), the base of the logarithm is 6, the result of the logarithm is 216, and the variable y is equivalent to the exponent in the exponential form.

Step 2: Apply Logarithmic to Exponential Conversion

Now that we have identified the base, result, and exponent, we can convert the logarithm to its equivalent exponential form using the property \(b^c = a\). Equating the terms we have identified to the conversion property gives us: \(6^y = 216\). This is the exponential form of \(\log _{6} 216=y\).

Key Concepts

Logarithmic Form

Logarithms are a unique way to express exponentiation in reverse. When we see a logarithm like \(\log _{6} 216 = y\), it tells us the exponent \(y\) to which the base \(6\) must be raised to get the number \(216\). In simple terms, a logarithm answers the question: "To what power must the base be raised to produce a given value?"

Logarithmic form is expressed as \(\log_b a = y\), where:

• \(b\) is the base of the logarithm
• \(a\) is the result or the number for which we are solving the logarithm
• \(y\) is the power or exponent that the base is raised to, resulting in \(a\)

Understanding logarithmic form helps in converting it to exponential form for easier calculation or comprehension.

Logarithm Properties

Understanding the properties of logarithms is crucial for simplifying and manipulating logarithmic expressions. Here are some key properties:

• Product Property: \(\log_b (mn) = \log_b m + \log_b n\), which combines the log of a product into the sum of logs.
• Quotient Property: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\), expressing the log of a quotient as a difference of logs.
• Power Property: \(\log_b (m^n) = n \cdot \log_b m\), which moves the exponent in the argument to the front as a multiplier.
• Change of Base Formula: \(\log_b a = \frac{\log_k a}{\log_k b}\), which allows conversion of logs to a different base \(k\).

These properties not only simplify complex logarithmic expressions but also aid in understanding their structure and behavior. Any logarithm expression can be manipulated using these properties to make computations easier.

Base of Logarithm

The base of a logarithm is a fundamental aspect because it determines the exponential growth factor. When we see an expression like \(\log_b a\), \(b\) is the base and it signifies the number that is repeatedly multiplied.

The base is crucial in conversions and calculations within both exponential and logarithmic forms:

• If \(b = 10\), it's a common logarithm, widely used in scientific and engineering contexts.
• If \(b = e\) (approximately 2.718), it's a natural logarithm, denoted as \(\ln\), prevalent in calculus and natural growth models.

In our problem, the base is \(6\), which means answering "to what power must 6 be raised to get 216?" This idea helps when switching between exponential and logarithmic formats and is essential in solving problems efficiently.