Logarithm Model Questions Set 1 Practice Questions Answers Test with Solutions & More Shortcuts

Answer: (c)

let $2^{40} = 10^x$

log $2^{40} = log 10^x$

40 $log^2 = x log^{10}$

40 (.301) = x

12.04 = x

i.e. x > 12

∴ x = 13

No. of digit in $2^{40}$ is 13

Answer: (d)

$\text"log"_7 \text"log"_5(√x + 5 + √x)$ = 0

use $\text"log"_a$ x = b⇒$a^b$=x

∴ $\text"log"_5 (√x + 5 + √x) = 7^0$ = 1

$√x + 5 + √ x = 5^1 = 5⇒2√x$ = 0

∴ x = 0

Answer: (a)

$\text"log"_{6} 216 √6 = \text"log"_6 (6)^3 (6)^{1/2}$

= $\text"log"_6 (6)^{7/2} = {7/2}{ \text"log"_6} 6 = 7/2$ (∵ $\text"log"_a$ a = 1)

Answer: (d)

$log 7^{25}$ = 25 log 7 = 25 × 0.845

= 21.125

Its characteristic is 21⇒No. of digits = 22

$log 8^{23} = log 2^{69}$ = 69 log 2

= 69 × 0.301 = 20.769

Its characteristic is 20⇒No. of digits = 21

$log 9^{20} = log 3^{40}$

= 40 × log3

= 40 × 0.477 = 19.08⇒No. of digits = 20

Near about is 22, 21, 20 so option (d) satisfy

Answer: (c)

${log_{13}(10)}/{log_{169}(10)} = {log_{13}(10)}/{log_{13^2}(10)}$

$(∵ log_{a^b} c = 1/b log_a c)$

= ${log_{13}10}/{1/2 log_{13}10} = 1/{1/2}$ = 2