Problem 86 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(\frac{1}{2} \log _{8}\left(x^{2}+5\right)-\log _{8}\left(x^{2}+5\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{8}\left((x^{2}+5)^{-1/2}\right) \).

Step 1: Apply the Power Rule

The power rule of logarithms states that the expression \( a \log_{b}(M) \) can be rewritten as \( \log_{b}(M^{a}) \). For the term \( \frac{1}{2} \log_{8}(x^{2}+5) \), we use the power rule to rewrite it as \( \log_{8}((x^{2}+5)^{1/2}) \).

Step 2: Apply the Quotient Rule

The quotient rule of logarithms states that \( \log_{b}(M) - \log_{b}(N) = \log_{b}\left(\frac{M}{N}\right) \). We can apply this rule to combine the two logarithms: \[ \log_{8}\left((x^{2}+5)^{1/2}\right) - \log_{8}(x^{2}+5) = \log_{8}\left(\frac{(x^{2}+5)^{1/2}}{x^{2}+5}\right) \].

Step 3: Simplify the Expression

Simplify the expression \( \frac{(x^{2}+5)^{1/2}}{x^{2}+5} \) by using the property \( (a^{b}) = \sqrt[b]{a} \). Here, \( (x^{2}+5)^{1/2} = \sqrt{x^{2}+5} \), leading to: \[ \frac{\sqrt{x^{2}+5}}{x^{2}+5} = \frac{1}{\sqrt{x^{2}+5}} \].

Step 4: Finalize the Logarithmic Expression

Now, substitute back into the logarithmic form: \[ \log_{8}\left(\frac{1}{\sqrt{x^{2}+5}}\right) = \log_{8}\left((x^{2}+5)^{-1/2}\right) \].

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a nifty tool that helps convert expressions of the form \( a\log_{b}(M) \) into more manageable pieces. Simply put, when you have a coefficient multiplying the logarithm, like \( \frac{1}{2}\log_{8}(x^{2}+5) \), it transforms into an exponent. This means you can rewrite it as \( \log_{8}((x^{2}+5)^{1/2}) \).
Think about the power rule as moving the multiplier inside the log function as an exponent. This trick allows complex expressions to become easier to handle, especially when combining multiple logarithmic terms.
Remember, practice makes perfect! The more you apply the power rule, the more intuitive it will become.

Quotient Rule of Logarithms

The quotient rule of logarithms is your go-to method when facing a subtraction between two logs with the same base, such as \( \log_{b}(M) - \log_{b}(N) \). According to this rule, this difference can be represented as a single log of a fraction: \( \log_{b}\left(\frac{M}{N}\right) \).
In our case, you have two logs: \( \log_{8}((x^{2}+5)^{1/2}) \) and \( \log_{8}(x^{2}+5) \). Using the quotient rule, these combine to become \( \log_{8}\left(\frac{(x^{2}+5)^{1/2}}{x^{2}+5}\right) \).
Applying the quotient rule is essential for simplifying logarithmic expressions, reducing complex problems into a compact and efficient form.

Simplifying Expressions

Simplifying expressions is about making them as concise and straightforward as possible. In logarithmic expressions, this simplifies the calculation and makes the expression easier to understand.
For example, consider \( \frac{\sqrt{x^{2}+5}}{x^{2}+5} \). Breaking it down further, the numerator’s square root is represented as an exponent of \( \frac{1}{2} \), and can rewrite this as \( (x^{2}+5)^{1/2} \). With this simplification, the fraction becomes \( \frac{1}{\sqrt{x^{2}+5}} \), aligning neatly with the logarithmic form \( \log_{8}\left((x^{2}+5)^{-1/2}\right) \).
Simplification not only makes logarithmic problems easier to solve, but also provides a clearer insight into the relationship between different components inside the equation.