Find the approximate value of _ 10 1005, given that _ 10 e = 0.43… - Vidyadip

Question

Find the approximate value of $\log_{10} 1005,$ given that $\log_{10} e = 0.4343.$

✓

Answer

Let:
$\text{y}=\text{f}(\text{x})=\log_{10}\text{x}$
here,
$x = 1000,$
$\text{x}+\triangle\text{x}=1005$
$\Rightarrow\triangle\text{x}=5$
$\Rightarrow\text{dx}=\triangle\text{x}=5$
For $x = 1000,$
$\text{y}=\log_{10}1000=\log_{10}(10)^3=3$
Now, $\text{y}=\log_{10}\text{x}=\frac{\log_\text{e}\text{x}}{\log_\text{e}10}$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{0.4343}{\text{x}}$
$\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}=1000}=\frac{0.4343}{1000}=0.0004343$
$\triangle\text{y}=\text{dy}=\frac{\text{dy}}{\text{dx}}\text{dx}=0.0004343\times5=0.0021715$
$\therefore\log_{10}1005=\text{y}+\triangle\text{y}=3.0021715$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate:
$\begin{vmatrix}\text{x}+\lambda&\text{x}&\text{x}\\\text{x}&\text{x}+\lambda&\text{x}\\\text{x}&\text{x}&\text{x}+\lambda\end{vmatrix}$

Using vectors, find the area of the triangle with vertices:

A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)
A(1, 2, 3), B(2, -1, 4) and C(4,5, -1)

Find the angle between the following pairs of lines:$\frac{\text{x}-5}{1}=\frac{2\text{y}+6}{-2}=\frac{\text{z}-3}{1}$ and $\frac{\text{x}-2}{3}=\frac{\text{y}+1}{4}=\frac{\text{z}-6}{5}$

Evaluate:
$\int\frac{\text{6x + 7}}{\sqrt{\text{(x - 5)(x - 4)}}}\text{dx}$.

Find a unit vector perpendicular to the plane A B C, where the coordinates of A, B and C are A (3, -1, 2), B (1, -1, -3) and C (4, -3, 1).

In a hospital, there are 20 kidney dialysis machines and that the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.

Evaluvate the following intregals:
$\int\frac{2\tan\text{x}+3}{3\tan\text{x}+4}\ \text{dx}$

One kind of cake requires 300 g of flour and 15 g of fat, another kind of cake requires 150 g of flour and 30 g of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600 g of fat, assuming that there is no shortage of the other ingradients used in making the cakes. Make it as an L.P.P. and solve it graphically.

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer’s profit on an item of model A is Rs. 15 and on an item of model B is Rs. 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Show that the function $\text{f}(\text{x})=\cot^{-1}(\sin\text{x}+\cos\text{x})$ is decreasing on $\Big(0,\frac{\pi}{4}\Big)$ and increasing on $\Big(\frac{\pi}{4},\frac{\pi}{2}\Big).$