Zoom In... Enhance... Now Zoom In Again

Converting $1000$ $\text{km}$ to $\text{cm}$ , we get $10^8 \text{cm}$ . Now, let us form a recurrence,

$\Rightarrow f(1) = 10^8$

Every time we zoom, we cut it by a factor of $2$ , that is, we divide the previous version by $2$ .

Therefore,

$\Rightarrow f(n) = \dfrac{f(n-1)}{2}$

The solution to this recurrence is $\Rightarrow f(n) = 390625 \times 2^{9-n}$

We want to find the value for $n$ such that $f(n) = 3$ .

Therefore, we can equate $390625 \times 2^{9-n}$ to $3$ .

Doing so, we get

$\Rightarrow 3 = 390625 \times 2^{9-n}$

or $n = 25.990$ . Thus we need to zoom in about $\boxed{25}$ times.

I think this might be the simpler way to solve this problem. You should at first calculate the natural logarithm values of $2$ and $33333333.3333$ in order to solve this. Also, you should know the formula $\log (m^n)=n \log m$ .

As the screen size remains the same and image becomes magnified double on each click, we can say that,

Since, here we see that on each click, the screen zooms by a factor of $2$ , so we can say that on clicking $x$ times, the screen zooms by a factor of $2^x$ . Now, the actual width at first is $1000\text{ km}=1000\times 10^5 \text{ cm}=10^8 \text{cm}$ . After zooming by $x$ clicks. we see a width of only $3\text{ cm}$ on the whole screen. So, from all this data, we can formulate an equation like this ---->

$\large \frac{10^8\text{ cm}}{2^x}=3\text{ cm}$

$\large \implies 2^x=\frac{10^8\text{ cm}}{3\text{ cm}}$

$\implies 2^x\approx 33333333.3333$

Applying logs on both sides (with base 'e', i.e., natural logarithm $\ln$ ), we have --->

$\implies \ln (2^x)\approx \ln (33333333.3333)$

$\implies x \ln(2)\approx 17.322$

$\implies x\times (0.693147)\approx 17.322$

$\implies x\approx \frac{17.322}{0.693147}$

$\implies x\approx \boxed{25}$

Thus, we need to click on the screen 25 times.

I think I have a good solution:

basically the 1000 km is multiplied by 0.5 each Time you click zoom in so clicking zoom in twice would be:

$1000\times 0.5\times 0.5$ which is the same as $1000\times 0.5^{2}$

So the width after clicking n times would be: $1000\times 0.5^{n}$

now 3 cm is the same as $3\times 10^{-5}$ Km

We set both equal to each other giving us:

$1000\times 0.5^{n} = 3\times 10^{-5}$

$0.5^{n} = 3\times 10^{-8}$

Now take logs to both sides: $\log {0.5 }^{n } = \log { 3\times {10}^{-8 } }$

$n\log { 0.5 } =\log { 3\times {10}^{-8 } }$

$n = \frac { \log { 3\times { 10 }^{ -8 } } }{ \log { 0.5 } }$

$n = 24.99$

$n = \boxed{25}$

That's it :D

Divide 1000 Km by 2( factor given) 3 times to get 125 Km,convert in meters by multiplying by 1000. Again divide by 2 ( 4 times to get 15625 mts,convert to cm by n multipling by 100. now divide by 2 ( 18 timesto get fig 3 cm). Thus,total 25 times is answer.

K.K.GARG,India

Use Mental Math and approximation technique and a bit of intuition

Lets start with an image of size 2cm, in order to zoom out to 1000km = 10^8cm we need to satisfy the relationship to determine n where n is the zoom factor

$2^n = 10^8$

$\Rightarrow nlog(2) = log(10^8) =8log(10)$ $\Rightarrow nlog(2) = 8$

We know $log(2) \approx 0.3$ so we get

$\Rightarrow n = \frac{8}{0.3} = \frac{80}{3} \approx 26.66$

The nearest answer is 25

$1000km=10^{8}cm$

We need to find n such that

$10^{8}/2^{n}=3$

Now

$2^{10}≈10^{3}$

$2^{20} * 100 ≈10^{8}$

$2^{20} * 32 * 3 ≈10^{8}$

$2^{25} * 3 ≈10^{8}$

Which is equivalent to our requirement and thus n=25

10^8=3X2^x 33333333.33=2^x Taking log both sides 7.522=xlog2 X=24.99