Properties of Logaritms

Properties of logarithms are following:

1. If x_1>0 and x_2>0 then color(red)(log_a(x_1x_2)=log_a(x_1)+log_a(x_2)) (logarithm of product equals sum of logarithms of factors). This property is true for any number of factors.

For example, log_3(60)=log_3(3*20)=log_3(3)+log_3(20)=1+log_3(20)=1+log_3(4*5)=1+log_3(4)+log_3(5).

2. If x_1>0 and x_2>0 then color(blue)(log_a(x_1/x_2)=log_a(x_1)-log_a(x_2)) (logarithm of quotient equals difference of logarithms of dividend and divisor).

For example, log_2(5/4)=log_2(5)-log_2(4)=log_2(5)-2.

3. If x_1<0 and x_2<0 then color(green)(log_a(x_1x_2)=log_a|x_1|+log_a|x_2|).
4. If x_1<0 and x_2<0 then color(165,42,42)(log_a(x_1/x_2)=log_a|x_1|-log_a|x_2|).
5. If x>0 then color(100,149,237)(log_a(x^r)=rlog_a(x)). For example, log_5(81)=log_5(3^4)=4log_5(3); log_3(sqrt(2))=log_3(2^(1/2))=1/2 log_3(2).

Following fact is true: if k is even number then log_a(x^k)=k log_a|x| for any x!=0.

For example, log_2(x^4)=4log_2|x|; log_3(x^2)=2log_3|x|.

6. If x>0 then color(218,165,32)(log_a(x)=(log_b(x))/(log_b(a))) (formula for changing base of logarithm).

For example, log_2(3)=(log_5(3))/(log_5(2)); log_a(b)=(log_b(b))/(log_b(a))=1/(log_b(a)).

7. If x>0 then color(148,0,211)(log_(a^k)(x^k)=log_a(x)).

For example, log_2(5)=log_(2^3)(5^3)=log_(sqrt(2))(sqrt(5)).

Example. Find log_5(6) if log_2(3)=a and log_2(10)=b.

Firs we since we are given logarithms with base 2, then we need to convert required logaritm to base 2:

log_5(6)=(log_2(6))/(log_2(5))=(log_2(2*3))/(log_2(10/2))=(log_2(2)+log_2(3))/(log_2(10)-log_2(2))=(1+a)/(b-1).