**Introduction to logaithms problems learning:**

In math, Learning Logarithms are very common function and it plays a vital role in problems involving variable quantities and exponents. Logarithms of a certain number represent the value which is multiplied by 10 for the number of times given into the logarithmic function. For example: Consider log6, it represents 1000000, similarly log 1000 represents 3. Real Numbers which are Positive will have real logarithms, negative & complex numbers will only contain complex logarithms. The commonly used base is 10, and if no base is mentioned, then it is considered as log with base 10.

There are five basic rulers to be followed and they are given away as follows,

1) Product Rule

Log _{a} x y = log _{a} x + log _{a} y

2) Quotient Rule

If log _{a} x / y = log _{a} x – log _{a} y

3) Power Rule

Log _{a} x ^{n} = n log _{a} x

4) Change of Base Rule

Log _{a} b = log _{c} b / log _{c} a

5) If a = b ^{y}, then y = log _{b} (a)

**Problem 1:**

Evaluate 2 log _{3} 6 + log _{3} 50 – log _{3} 1800

**Sol:**

2 log _{3} 6 + log _{3} 50 – log _{3} 1800

= log _{3} 6 ^{2} + log _{3} 50 – log _{3} 1800

= log _{3} 36 + log _{3} 50 – log _{3} 1800

= log _{3} ((36 * 50) / 1800)

= log _{3} 1

= 0

**Problem 2:**

Evaluate log _{5} 25 + log _{3} 243 – 2 log _{4} 2

**Sol:**

log_{5} 25 + log _{3} 243 – 2 log _{4} 2

= log _{5} 5 ^{2} + log _{3} 3 ^{5} – log _{4} 2
^{2}

= 2 log _{5} 5 + 5 log _{3} 3 – log _{4} 4

= 2(1) + 5(1) – (1)

= 2+5-1

= 7-1

=6

**Problem 3:**

Evaluate 2 log _{3} 27 + log _{3} 81 – 3 log _{3} 9

**Sol:**

=2 log _{3} 3 ^{3} + log _{3} 3 ^{4} – 3 log _{3} 3 ^{2}

= 3 * 2 log _{3} 3 + 4 log _{3} 3 – 2 * 3 log _{3} 3

= 6 log _{3} 3 +4 log _{3} 3– 6 log _{3} 3

=6(1) + 4(1) -6(1)

=10-6

=4