Logarithms: Definition, Five Laws, Change of Base and Counting Digits with log 2

Logarithms: Turning Powers Into Simple Arithmetic

Before calculators existed, a logarithm turned long multiplications into easy additions. Today it survives in exams because it is a compact way of asking one question: to what power must the base be raised to give this number? Since 2 raised to the power 5 gives 32, the logarithm of 32 to the base 2 is 5.

Logarithm questions in SSC, Railway and teacher-recruitment papers are short, and almost every one yields to three tools: converting between log form and power form, five standard laws, and the value of log 2. Learn these and a typical question takes under thirty seconds.

The next cards give the definition, the laws, worked examples with real numbers, and the traps that quietly cost marks.

1. What is the value of log_2 32?

2. What is the value of log10 25 + log10 4?

3. If log_x 81 = 4, what is the value of x?

4. What is the value of log_9 27?

5. Given log 2 = 0.3010, what is log 5 (base 10)?

6. Given log 2 = 0.3010, how many digits does 2^20 have?

7. Which of the following statements is true for positive a, b, and N?

8. What is the value of 5^(log_5 12)?