Question

Match the expression with the logarithm that has the same value. Justify your answer. A. \(\log _3 64\) B. \(\log _3 3\) C. \(\log _3 12\) D. \(\log _3 36\) $$2 \log _3 6$$

Short answer

The logarithmic expression \(2 \log _3 6\) has the same value as D, which is \(\log _3 36\).

Step-by-step solution

Understand the Given Log

The given is a transformed logarithm, which means that it has been changed in some way from the original log. Here, the transformation is multiplication of the logarithm \(\log _3 6\) by 2. We need to simplify or expand this expression and compare it with the available options A, B, C, and D.

Simplify the Log

We need to apply the power rule of logarithms, which states: given a logarithm \(\log_a b^c\), it can be rewritten as \(c \log_a b\). Therefore, \(2 \log _3 6\) simplifies to \(\log _3 6^2\) or \(\log _3 36\)

Match the Simplified Expression

Now, match this simplified expression \(\log _3 36\) with the options given in the problem statement. As we can see, \(\log _3 36\) is same as option D. So, D (\(\log _3 36\)) is the correct match to the given logarithmic expression.

Key concepts

Logarithmic Expression

A logarithmic expression is a mathematical expression involving the logarithm function. Logarithms are the inverse operations of exponentiation. In simpler terms, while an exponentiation determines what power to raise a number to get another number, a logarithm tells us what power to raise a base to get that number.
For a basic understanding:

• Consider the expression \ \( \log_b a \ \). Here, \ \( b \ \) is the base of the logarithm, and \ \( a \ \) is the number which the logarithm is being applied to.
• The value of the logarithm is what power you raise \ \( b \ \) to get \ \( a \ \).

In the context of our exercise, \ \( \log_3 36 \ \) indicates that we are asking "what power do we need to raise 3 to, in order to get 36?"
Understanding how logarithmic expressions are structured will aid in simplifying them and comparing their values.

Power Rule of Logarithms

The power rule of logarithms is a fundamental rule that is extremely useful for transforming logarithmic expressions. This can make expressions easier to manipulate or solve.
The power rule states that \ \( \log_a b^c = c \log_a b \ \). This means that when you have a power inside a logarithm, you can bring it in front of the log, effectively transforming the expression.
For instance:

• If we have \ \( \log_3 6^2 \ \), applying the power rule gives us \ \( 2 \log_3 6 \ \). Conversely, if we start with \ \( 2 \log_3 6 \ \), we can apply the rule in reverse to get back to \ \( \log_3 6^2 \ \).

The exercise uses this principle, transforming \ \( 2 \log_3 6 \ \) into \ \( \log_3 36 \ \) by recognizing that \ \( 6^2 = 36 \ \). This step is crucial for simplifying expressions and is necessary for matching them to others that might initially look different.

Simplifying Logarithms

Simplifying logarithms involves reducing a logarithmic expression to its simplest form, enabling easier comparison and integration with other expressions. This process often involves applying rules such as the power, product, and quotient rules of logarithms.
Simplifying makes expressions manageable:

• In our exercise, we have \ \( 2 \log_3 6 \ \). By applying the power rule, we simplify it to \ \( \log_3 36 \ \).
• This simplification allows us to compare it directly with other options given in the exercise, such as \ \( \log_3 36 \ \), which is option D.

During simplification, knowing when and how to apply logarithm rules is key. These rules help transform expressions into equivalent forms that are easier to interpret or match with other expressions. Mastery of these techniques hones one's problem-solving skills and facilitates more efficient computations involving logarithms.