Question

Find the value of $ \sqrt[5]{{.00000165}} $ , given $ \log 165 = 2.2174839 $ , $ \log 697424 = 5.8434968 $ .

Answer 1

Hint: We have to find the fifth root of the number $ 0.00000165 $ which would be a decimal number which when multiplied by itself $ 5 $ times would give the original number. Such questions can be solved using a logarithmic function. Logarithmic function is defined as,
if $ {a^y} = x $ , then $ {\log _a}x = y $ .
Without any base given, we assume the base to be $ 10 $ . We will also use some properties of logarithmic function. Some values are provided in the question which is to be used while solving.

Complete step by step solution:
We have to find the fifth root of the number $ 0.00000165 $
Let us assume $ \sqrt[5]{{.00000165}} = x $
Then we can raise power $ 5 $ on both sides and simplify as follows,
 $
  {\left( {\sqrt[5]{{.00000165}}} \right)^5} = {x^5} \\
   \Rightarrow 0.00000165 = {x^5} \\
   \Rightarrow \dfrac{{165}}{{100000000}} = {x^5} \\
   \Rightarrow 165 \times {10^{ - 8}} = {x^5} \;
  $
Now we take $ \log $ on both sides. Taking $ \log $ means putting both the sides under logarithmic function.
 $ \Rightarrow \log \left( {165 \times {{10}^{ - 8}}} \right) = \log \left( {{x^5}} \right) $
Here we will use a property of log function given as,
 $ \log (a \times b) = \log a + \log b $
Thus, we can write,
 $
  \log \left( {165 \times {{10}^{ - 8}}} \right) = \log \left( {165} \right) + \log \left( {{{10}^{ - 8}}} \right) \\
   \Rightarrow \log \left( {165} \right) + \log \left( {{{10}^{ - 8}}} \right) = \log \left( {{x^5}} \right) \;
  $
Here again we will use a property of log function given as,
 $ \log ({a^b}) = b\log a $
Thus,
 $
  \log \left( {{{10}^{ - 8}}} \right) = - 8\log 10 \\
  \log \left( {{x^5}} \right) = 5\log x \;
  $
Thus we can write,
 $
  \log \left( {165} \right) + \log \left( {{{10}^{ - 8}}} \right) = \log \left( {{x^5}} \right) \\
   \Rightarrow \log \left( {165} \right) - 8\log 10 = 5\log x \;
  $
We have been given the value $ \log 165 = 2.2174839 $
Also, we know from basic logarithmic property, $ \log 10 = 1 $
\[
   \Rightarrow 2.2174839 - 8 = 5\log \left( x \right) \\
   \Rightarrow 5\log x = - 5.7825161 \\
   \Rightarrow \log x = \dfrac{{ - 5.7825161}}{5} = - 1.15650322 \;
 \]
In the question we have been given the value of $ \log 697424 = 5.8434968 $ . We will try to use this value to find the value of $ x $ .
We will add $ 7 $ to both sides of the equation.
\[
   \Rightarrow \log x + 7 = - 1.15650322 + 7 \\
   \Rightarrow \log x + \log {10^7} = 5.8434968 \\
   \Rightarrow \log \left( {{{10}^7}x} \right) = \log 5.8434968 \\
   \Rightarrow {10^7}x = 697424 \\
   \Rightarrow x = 0.0697424 \;
 \]
Thus, we get the value of $ x $ as \[0.0697424\]
Hence, $ \sqrt[5]{{.00000165}} = 0.0697424 $
So, the correct answer is “0.0697424”.

Note: We use the properties of the logarithmic function to find the value of fifth root of the given decimal number. We could have also calculated the value of $ x $ as $ antilog\left( { - 1.15650322} \right) $ . While solving a problem it is important to take note of the information given and use them in the solution. We can also check the solution as by multiplying the result by itself $ 5 $ times it should yield the original number, i.e. \[{\left( {0.0697424} \right)^5} = 0.00000165\].