Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{8}\left(\frac{64}{\sqrt{x+1}}\right)$$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \( \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) \) is \( 2 - \frac{1}{2} \cdot \log _{8} (x+1) \)

Step 1: Expressing as two separate logarithms

First change the fraction into two separate logarithms according to the property \( \log_b(a / c) = \log_b(a) - \log_b(c) \). This is done as follows: \( \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) = \log _{8}(64) - \log _{8}\left(\sqrt{x+1}\right) \)

Step 2: Simplify log base 8 of 64

In the next step, simplify the part with 64. Since \( 8^2 = 64 \), it follows that \( \log _{8}(64) = 2 \). So the equation becomes: \( 2 - \log _{8}\left(\sqrt{x+1}\right) \)

Step 3: Denote the square root as an exponent

The square root is denoted as an exponent of 1/2. So, \( \sqrt{x+1} \) equals \( (x+1)^{1/2} \). The equation now looks like this: \( 2 - \log _{8}\left((x+1)^{1/2}\right) \)

Step 4: Apply exponent rule

Next, utilize the property of logarithms \( \log_b{a^c} = c \cdot \log_b(a) \) and apply it to our equation. The final expanded form will look like this: \( 2 - \frac{1}{2} \cdot \log _{8} (x+1) \)