Question

Find the exact value of the logarithmic expression without using a calculator.\(\log _{9} \frac{1}{18}\)

Short answer

-1/2 log(2) - 1

Step-by-step solution

Apply the Reciprocal Property of Logarithms

The logarithm of the reciprocal of a number is the negative of the logarithm of that number, so \(\log _{9} \frac{1}{18} = -\log_{9}(18)\).

Apply the Change-of-Base Formula

The change-of-base formula allows you to rewrite a logarithm in terms of logarithms with a different base. In this case, let's change it to a base 3 logarithm. \(-\log_{9}(18) = -\frac{\log_{3}(18)}{\log_{3}(9)}\).

Simplify Known Values

Both 9 and 18 can be expressed as powers of 3, which simplifies the logarithm. \(9 = 3^2\) and \(18 = 2*3^2\). Hence, \(-\frac{\log_{3}(18)}{\log_{3}(9)} = -\frac{\log_{3}(2*3^2)}{\log_{3}(3^2)} = -\frac{\log_{3}(2) + 2\log_{3}(3)}{2\log_{3}(3)} = -\frac{1}{2}\log_{3}(2) - 1\).

Summarize the result

Therefore, \(\log _{9} \frac{1}{18} = -\frac{1}{2}\log_{3}(2) - 1\).

Key concepts

Reciprocal Property of Logarithms

Logarithms have fascinating properties that simplify calculations, and one of these is the reciprocal property. This property states that the logarithm of the reciprocal of a number is the negative of the logarithm of that number itself. In simpler terms, if you have an expression like \( \log_b \left( \frac{1}{x} \right) \), you can rewrite it as \( -\log_b(x) \). This is particularly useful when dealing with fractions, as it allows us to convert a division operation into a subtraction operation.
To see this property at work, let's look at the expression \( \log_9 \left( \frac{1}{18} \right) \) from the original problem. Using the reciprocal property of logarithms, we transform this expression to \( -\log_9(18) \), simplifying our problem considerably. This step helps us move forward in solving without getting bogged down by fractions.

Change-of-Base Formula

The change-of-base formula is another crucial tool in the logarithmic toolkit, allowing us to evaluate logarithms with bases that are not readily computable by hand. It states that any logarithm \( \log_b(x) \) can be rewritten using two different bases. This formula is particularly useful when working with calculator-unfriendly bases.
For instance, in our problem, to compute \( -\log_9(18) \), we can utilize the change-of-base formula: \( -\log_9(18) = -\frac{\log_3(18)}{\log_3(9)} \). By expressing the logarithms with base 3, the problem becomes much simpler, as both 9 and 18 can be conveniently expressed using powers of 3. This step is pivotal in simplifying any further calculations required to evaluate the expression completely.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves breaking down numbers into their prime factors and recognizing familiar forms. In our case, both 9 and 18 can be expressed in terms of powers of 3: \( 9 = 3^2 \) and \( 18 = 2 \times 3^2 \).
When we apply these transformations using the change-of-base formula, we find that \(-\frac{\log_3(18)}{\log_3(9)}\) becomes a simpler expression. By substituting, it turns into \(-\frac{\log_3(2) + 2\log_3(3)}{2\log_3(3)}\). Breaking it down further, this simplifies to \(-\frac{1}{2}\log_3(2) - 1\).
Leveraging these skills in recognizing numeric patterns and their log identities allows for clean and clear simplifications of what might initially seem complex expressions, helping with both understanding and calculation.