Log basics

Algebra Level 3

You are given that $\log_{10} 5 \approx 0.6990$ is accurate to 3 decimal places. Find the number of digits in the number $50^{50}$ .

Note: For accurate results via approximation, you have to verify that you have enough significant figures.

The answer is 85.

2 solutions

Aniket Sanghi
Feb 2, 2016

find 50*log50 = 84.95 this implies that no. of digits is 85....i.e.characteristic+1

Moderator note:

Simple standard approach.

In the question, for clarity, it should be clearly stated that the value is accurate to 3 sig fig. Otherwise, we could have used a slightly different approximation, and obtained a wrong answer.

Marvelous!

Heder Oliveira Dias - 5 years, 4 months ago

Thank you...............

Aniket Sanghi - 5 years, 4 months ago

Arulx Z
Feb 3, 2016

Number of digits in $n$ is

$\left\lfloor \log { n } \right\rfloor +1$

Plugging in $50^{50}$ , length is

$=\left\lfloor \log { { 50 }^{ 50 } } \right\rfloor +1\\ =\left\lfloor 50\log { 50 } \right\rfloor +1$

Since $50=5 \cdot 10$ ,

$=\left\lfloor 50\left( \log { 5 } +\log { 10 } \right) \right\rfloor +1\\ =\left\lfloor 50\left( 0.699+1 \right) \right\rfloor +1\\ =\left\lfloor 84.95 \right\rfloor +1\\ =85$

Log basics

The answer is 85.

2 solutions

Moderator note:

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