Tips And Tricks And Shortcut For Logarithm

Question 1 Which of the following statement is not correct?

Option

A) log (1 + 2 + 3) = log 1 + log 2 + log 3

B) log( 2+3) = log (2×3)

C) log₁₀ 1 = 0

D) log₁₀ 10 = 1

Solution:   

a)  log( 1+2+3) = log6 = log (1×2×3) = log1+ log2+log3

b) log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3

log (2 + 3) ≠ log (2 x 3)

c) Since,  log_a 1=0,
so log₁₀ 1=0

d) Since, log_a a = 1
so, log₁₀ 10 = 1.

Correct Answer : B

Question 2 If X is an integer then solve(log₂ X)²– log₂x⁴-32 = 0

Solution :    Let us consider,(log₂ X)²– log₂x⁴-32 = 0 —— equation 1

let log₂ x= y

equation 1 = y²– 4y-32=0

y²-8y+ 4y – 32= 0

y(y-8) + 4(y-8) =0

(y-8) (y+4)= 0

y=8, y= -4

log₂X=8 or, log₂X = -4

X= 2⁸ = 256, 0r 2^-4 = \frac{1}{16} \

Since, X is an integer so X = 256.

Question 3 Solve for x: \log_{2}(x + 3) + log_{2}(x – 1) = 3

Solution: Combine the logarithms using the logarithm property log(a) + log(b) = log(a * b):
\log_{2}((x+3)\times(x-1)) = 3

2^3=(x+3)\times(x-1)

8=(x+3)\times(x-1)

8=x^2+2x-3

x^2+2x-11=0

(x+4)(x-2)=0

x = -4,2