Problem 70
Question
Simplify. $$ \log _{6} 6^{2} $$
Step-by-Step Solution
Verified Answer
The expression simplifies to 2.
1Step 1: Understand the Expression
The given expression is \( \log_{6} 6^2 \), which means the logarithm base 6 of \( 6^2 \). The goal is to simplify this expression.
2Step 2: Use Logarithmic Power Rule
According to the logarithmic power rule, \( \log_b (a^c) = c \times \log_b a \). Applying this rule to the expression \( \log_{6} 6^2 \), we can rewrite it as \( 2 \times \log_{6} 6 \).
3Step 3: Evaluate the Basic Logarithm
Next, evaluate \( \log_{6} 6 \). The logarithm of a number to its own base is 1, so \( \log_{6} 6 = 1 \).
4Step 4: Simplify Using Step 3 Result
Substitute \( \log_{6} 6 = 1 \) back into our expression: \( 2 \times 1 = 2 \). Thus, the simplified form of \( \log_{6} 6^2 \) is 2.
Key Concepts
Logarithmic Power RuleLogarithm BaseBasic Logarithm Properties
Logarithmic Power Rule
Understanding the logarithmic power rule is key to simplifying expressions where a logarithm is raised to a power. The rule states that if you have an expression like \( \log_b (a^c) \), it is equivalent to \( c \times \log_b a \). This means that you can "move" the exponent in front of the logarithm, essentially multiplying it. This rule is extremely useful when dealing with complex logarithmic expressions.
In the case of \( \log_{6} 6^2 \), the exponent here is 2. By applying the logarithmic power rule, we can rewrite the expression as \( 2 \times \log_{6} 6 \). This simplification allows us to perform further calculations more easily.
In the case of \( \log_{6} 6^2 \), the exponent here is 2. By applying the logarithmic power rule, we can rewrite the expression as \( 2 \times \log_{6} 6 \). This simplification allows us to perform further calculations more easily.
Logarithm Base
The base of a logarithm is a critical component. It determines which number you need to raise to a certain power to get the argument of the logarithm. In a logarithmic expression like \( \log_b a \), \( b \) is the base. A deeper understanding of the base can help demystify the functioning of logarithms.
In the expression \( \log_{6} 6^2 \), the base is 6. This implies that the logarithm is finding out what power 6 has to be raised to in order to equal \( 6^2 \). This base-specificity explains why using basic logarithm properties can greatly simplify computations. Generally, when the base matches the number being operated on, simplifications become much easier.
In the expression \( \log_{6} 6^2 \), the base is 6. This implies that the logarithm is finding out what power 6 has to be raised to in order to equal \( 6^2 \). This base-specificity explains why using basic logarithm properties can greatly simplify computations. Generally, when the base matches the number being operated on, simplifications become much easier.
Basic Logarithm Properties
Basic properties of logarithms are fundamental tools that can simplify solving problems swiftly. One of the most essential properties is that \( \log_b b = 1 \), where \( b \) is the base. This property is because a number raised to the power of 1 is itself.
Applying this to our example: for \( \log_{6} 6 \), since the base \( 6 \) and the argument \( 6 \) are identical, the result is 1. This means \( \log_{6} 6 = 1 \).
Armed with this knowledge, the computation becomes easy. Considering \( 2 \times \log_{6} 6 \), we substitute \( \log_{6} 6 = 1 \), leading us directly to \( 2 \times 1 = 2 \). This illustrates how understanding and utilizing basic logarithm properties can simplify even the complex-looking expressions dramatically.
Applying this to our example: for \( \log_{6} 6 \), since the base \( 6 \) and the argument \( 6 \) are identical, the result is 1. This means \( \log_{6} 6 = 1 \).
Armed with this knowledge, the computation becomes easy. Considering \( 2 \times \log_{6} 6 \), we substitute \( \log_{6} 6 = 1 \), leading us directly to \( 2 \times 1 = 2 \). This illustrates how understanding and utilizing basic logarithm properties can simplify even the complex-looking expressions dramatically.
Other exercises in this chapter
Problem 69
Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x} $$
View solution Problem 69
Evaluate each expression. $$ \log _{7} 7^{2} $$
View solution Problem 70
Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{2 x} $$
View solution Problem 70
Evaluate each expression. $$ \log _{7} \sqrt{7} $$
View solution