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Question: Find the value of \( \sqrt[5]{{.00000165}} \) , given \( \log 165 = 2.2174839 \) , \( \log 697424 = ...

Find the value of .000001655\sqrt[5]{{.00000165}} , given log⁡165=2.2174839\log 165 = 2.2174839 , log⁡697424=5.8434968\log 697424 = 5.8434968 .

Explanation

Solution

Hint : We have to find the fifth root of the number 0.000001650.00000165 which would be a decimal number which when multiplied by itself 55 times would give the original number. Such questions can be solved using a logarithmic function. Logarithmic function is defined as,
if ay=x{a^y} = x , then log⁡ax=y{\log _a}x = y .
Without any base given, we assume the base to be 1010 . 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 0.000001650.00000165
Let us assume .000001655=x\sqrt[5]{{.00000165}} = x
Then we can raise power 55 on both sides and simplify as follows,
(.000001655)5=x5 ⇒0.00000165=x5 ⇒165100000000=x5 ⇒165×10−8=x5   {\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⁡\log on both sides. Taking log⁡\log means putting both the sides under logarithmic function.
⇒log⁡(165×10−8)=log⁡(x5)\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×b)=log⁡a+log⁡b\log (a \times b) = \log a + \log b
Thus, we can write,
log⁡(165×10−8)=log⁡(165)+log⁡(10−8) ⇒log⁡(165)+log⁡(10−8)=log⁡(x5)   \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⁡(ab)=blog⁡a\log ({a^b}) = b\log a
Thus,
log⁡(10−8)=−8log⁡10 log⁡(x5)=5log⁡x   \log \left( {{{10}^{ - 8}}} \right) = - 8\log 10 \\\ \log \left( {{x^5}} \right) = 5\log x \;
Thus we can write,
log⁡(165)+log⁡(10−8)=log⁡(x5) ⇒log⁡(165)−8log⁡10=5log⁡x   \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\log 165 = 2.2174839
Also, we know from basic logarithmic property, log⁡10=1\log 10 = 1

⇒2.2174839−8=5log⁡(x) ⇒5log⁡x=−5.7825161 ⇒log⁡x=−5.78251615=−1.15650322   \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\log 697424 = 5.8434968 . We will try to use this value to find the value of xx .
We will add 77 to both sides of the equation.

⇒log⁡x+7=−1.15650322+7 ⇒log⁡x+log⁡107=5.8434968 ⇒log⁡(107x)=log⁡5.8434968 ⇒107x=697424 ⇒x=0.0697424   \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 xx as 0.06974240.0697424
Hence, .000001655=0.0697424\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 xx as antilog(−1.15650322)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 55 times it should yield the original number, i.e. (0.0697424)5=0.00000165{\left( {0.0697424} \right)^5} = 0.00000165.