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)`.