Problem 9 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in exponential form. Do not use a calculator. $$\log _{6} 36=2$$

Step-by-Step Solution

Answer

\( 6^2 = 36 \)

Step 1: Understand the Properties of Logarithms

The given statement is a logarithmic equation, \( \log_{6}36=2 \). It tells you that the logarithm of 36 with base 6 equals 2, meaning that when the base 6 is raised to the power of 2, it equals 36.

Step 2: Convert the Logarithmic Form to Exponential Form

To convert the logarithmic form \( \log_{6}36=2 \) to exponential form, interpret the equation. The base of the logarithm (6) becomes the base of the power, the result of the logarithm (2) becomes the exponent, and the number (36) becomes the result of the exponential expression. Therefore, the equivalent exponential form is \( 6^2 = 36 \).

Key Concepts

exponential form

Exponential form is a way to express numbers using a base and an exponent. This mathematical notation simplifies repeated multiplication. For example, instead of writing 6 multiplied by itself, we write it as \( 6^2 \). In this form, 6 is the base and 2 is the exponent. Understanding the conversion between logarithmic and exponential forms is crucial. A logarithmic equation like \( \log_{6}36=2 \) tells us that 36 is the number obtained when 6, the base, is raised to the power of 2. So, in exponential form, this is expressed as \( 6^2 = 36 \), emphasizing the relationship between the base, the exponent, and the resulting value.

properties of logarithms

Logarithms have several properties that simplify complex mathematical problems. Knowing these can deepen your understanding and help with solving logarithms efficiently.

• **Product Property**: \( \log_b (XY) = \log_b X + \log_b Y \).
• **Quotient Property**: \( \log_b \left( \frac{X}{Y} \right) = \log_b X - \log_b Y \).
• **Power Property**: \( \log_b (X^k) = k \log_b X \). Here, the exponent becomes a multiplier.
• **Change of Base Formula**: You can change the base of a logarithm using \( \log_b X = \frac{\log_c X}{\log_c b} \).

These properties form the backbone of logarithmic calculations. They provide flexibility in managing expressions by allowing complex ones to be broken down into simpler parts.

logarithmic equations

Logarithmic equations are equations that involve a logarithm with a variable. Solving these equations usually involves applying properties of logarithms to isolate the variable.To solve a logarithmic equation, you might:

• Use the properties of logarithms to simplify the equation.
• Convert the logarithm into its exponential form, as we did with \( \log_{6}36=2 \), leading to \( 6^2 = 36 \). This step often makes the variable more approachable.
• Open up expressions with known values and arrange terms to isolate the variable.

Finally, once you have isolated the variable, it's important to check your solution. Verification ensures there aren't any undefined log terms or other errors in the calculation. By practicing these steps, solving logarithmic equations becomes more intuitive and manageable.