Question

If $log_{10} 2 = 0.3010$, $log_{10} 3 = 0.4771$, then number of digits in 4⁸×3⁷×5³ is 'P'. What is the value of $\frac{P-1}{2}$ is?

Answer 1

Answer:

Let 4^8 * 3^7 * 5^3  = X

taking log to the base 10 on both the sides,      

(consider log (y) to be log to the base 10 and not the natural log)

log 5  = log (10/2) = log 10 - log 2 = 1 - 0.301 = 0.699

so,

log (4^8 * 3^7 * 5^3 )   =   log X

using log property

8 log 4 + 7 log 3 + 3 log 5  =  log X

or,

16 log 2 + 7 log 3 + 3 log 5 = log X

substitute the values,

16(0.301) + 7(0.4771) + 3(0.699) = log X

4.816 + 3.339 + 2.097 = log X

10.252 = log x

or , 10 ^ 10.25 = X

so the number of digits in x will be (10 ^10)  = 11

so p = 11

and hence the desired result will be 11 - 1 / 2 = 5

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Answer 2

Answer:

(P-1)/2 = 5

Step-by-step explanation:

Let N = 4^8 * 3^7 * 5^3 = (2^16) * (3^7) * (10/2)^3 = (2^13) * (3^7) * (10^3)

Taking log base 10 on each side, we get  logN = 13*log2 + 7log3 + 3log10

Or, logN = 13*0.3010 + 7*0.4771 + 3*1.0000

Or, logN = 3.9130 + 3.3397 + 3.0000 = 10.2527

Therefore N = 10^10.2527 and N would have 11 digits.

Therefore, P = 11

and (P-1)/2 = (11-1)/2 =5

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