Problem 118 - Introductory and Intermediate Algebra for College Students 4th

Question

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

Step-by-Step Solution

Answer

The exact value of \( \log_{4}\left[\log_{3}\left(\log_{2} 8\right)\right] \) is 0.

Step 1: Simplify Innermost Logarithm

Start by simplifying the innermost logarithm: \( \log_{2}{8} \). Using the property \( log_b(b^x) = x \), \( \log_{2}{2^3} = 3 \) because 8 is \( 2^3 \). So, \( \log_{2}{8} = 3 \).

Step 2: Simplify Next Inner Logarithm

Next, we simplify the next logarithm: \( \log_{3}{3} \). Following from the first step, we replace the value we found into this expression, which gives us \( \log_{3}{3} \). Using the same property, \( \log_{3}{3^1} = 1 \). So, \( \log_{3}{3} = 1 \).

Step 3: Simplify Outer Logarithm

Finally, we simplify the outermost logarithm: \( \log_{4}{1} \). We replace the value we found in the previous step into this expression which gives us \( \log_{4}{1} \). Since any number to the power of 0 is 1, we could rewrite 1 as \( 4^0 \), hence using the logarithm property, \( \log_{4}{4^0} = 0 \). So, \( \log_{4}{1} = 0 \).