Question

If \[{{\log }_{e}}5=1.609\] , then the approximate value of \[\log 5.1\] is?
1. \[1.611\]
2. \[1.701\]
3. \[1.809\]
4. \[1.629\]

Answer 1

Hint: We can solve this question with the basic idea that the derivative of a logarithmic function can be said to be the reciprocal of the things inside it. Now when we use the properties of logarithms we can make the differentiation process easier. Now we know that the derivative of logarithmic function is reciprocal of the argument and we multiply it with the arguments derivative. We can express all of this by
\[f(x)=\log (u)\]
\[f'(x)=\dfrac{1}{u}\dfrac{d\log (x)}{dx}\]

Complete step by step answer:
Now here in this question it is already given making us know that \[\log 5=1.609\]
Now using this information we need to find the approximate value of \[\log 5.1\]
Now to find the value of x here and the dx here. Since the value we know already is \[\log 5=1.609\] we can let \[x=5\] therefore we can now find the value of dx using
\[x+dx=5.1\]
Substituting and subtracting we get that
\[dx=0.1\]
Now to solve this further we will assume that \[y=\log x\]
Now on differentiating this we get that \[\dfrac{dy}{dx}=\dfrac{1}{x}\]
Now we know that \[x=5\] so finding its differentiate in the function we get \[f'(x)=\dfrac{dy}{dx}=\dfrac{1}{5}\]
Now to express we know that the value of \[f(x+\vartriangle x)=\log (5.1)\]
Now to find \[f(x+\vartriangle x)=\log (5.1)\] approximate value
\[f(x+\vartriangle x)=\log (5+0.1)\]
Differentiating the equation \[f(x+\vartriangle x)=\log (5.1)\] with respect to x
The derivative of f(x) can be given by
     \[f'(x)=\underset{\vartriangle x\to 0}{\mathop{\lim }}\,\dfrac{f(x+\vartriangle x)-f(x)}{\vartriangle x}\]
We can write this as
\[f(x+\vartriangle x)=\vartriangle xf'(x)+f(x)\]
Substituting we get
\[f(5.1)=0.1f'(5)+f(5)\]
Now putting the value of f(x) and its derivative we get that
\[f(5.1)=0.1\times \dfrac{1}{5}+\log 5\]
Now we also know the value of log function since its given in the question which is \[{{\log }_{e}}5=1.609\] .Substituting this value in equation we get
\[f(5.1)=0.1\times \dfrac{1}{5}+1.609\]
Therefore \[f(5.1)=1.611\] . This is the approximate value needed.

So, the correct answer is “Option 1”.

Note: We can only solve certain logarithmic differentiation using logarithmic Laws. Using this method we can find the values of many other values of x which are close to a previously known value we usually cannot use the laws.