Problem 39 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right] $$

Step-by-Step Solution

Answer

Therefore, the expanded expression becomes \(2 + 2 \log x + \frac{1}{3} \log (1-x) - 1.6902 - 2 \log (x+1)\)

Step 1: Decompose multiplication inside the logarithm

Using the property \(\log a*b = \log a + \log b\), break down the \(\log\) of the multiplication into separate terms. \[\log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right] = \log (10x^2) + \log(\sqrt[3]{1-x}) - \log(7(x+1)^2)\]

Step 2: Bring out exponents as multipliers

Utilize the property \(\log a^n = n \log a\) to handle the exponents inside the logarithms:\[= 2 \log (10x) + \frac{1}{3} \log (1-x) - 2 \log (7x+7)\]

Step 3: Further decompose logarithms

Continue decomposing the terms using multiplication and addition properties of logarithm:\[= 2[\log 10 + \log x]+ \frac{1}{3} \log (1-x) - 2 [\log 7 + \log (x+1)]\]

Step 4: Evaluate numerical logarithms

Expressions like \(\log 10\) and \(\log 7\) can be resolved as they are logarithmic expressions of pure numbers:\[= 2[\log 10 + \log x]+ \frac{1}{3} \log (1-x) - 2 [\log 7 + \log (x+1)]= 2[1+ \log x]+ \frac{1}{3} \log (1-x) - 2 [0.8451 + \log (x+1)]\]