Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Common and Natural Logarithms and Change of Base

Updated 1 October 2026 · Fact-checked

A common logarithm has base 10 (log x). A natural logarithm has base e ≈ 2.718 (ln x). The change of base formula, log_b a = log a ÷ log b, converts any base into one you know. Use it to simplify, compare or solve mixed-base logarithm MCQs.

Understand Common and Natural Logarithms and Change of Base

A logarithm answers one question: to what power must the base be raised to get a number? If 10² = 100, then log base 10 of 100 is 2.

A common logarithm uses base 10. It is written log x. When a CA Foundation question gives log with no base, it usually means base 10 unless the question says otherwise. Common logs suit calculations because our number system is base 10.

A natural logarithm uses base e, where e is a constant about 2.718. It is written ln x. It appears in growth and decay, continuous compounding and calculus. Since e^1 = e, ln e = 1. Also ln 1 = 0, as with every base.

Different bases cause trouble when one question mixes them. The change of base formula fixes this. It lets you rewrite a logarithm in any base you like. You can then use the usual laws of logarithms to simplify.

The two bases are linked: ln x = 2.3026 × log x (approximately), because ln 10 ≈ 2.3026. You rarely need this in MCQs unless the question gives you values.

Key formulas to remember

Definition: log_b a = x ⇔ b^x = a; Needs a > 0, b > 0 and b ≠ 1.
Common logarithm: log x means log_10 x; Base 10. Then log 10 = 1 and log 1 = 0.
Natural logarithm: ln x means log_e x; Base e ≈ 2.718. ln e = 1 and ln 1 = 0.
Change of base: log_b a = log_c a ÷ log_c b; c is any valid new base. Common choices are 10 or e.
Reciprocal rule: log_b a = 1 ÷ log_a b; So log_b a × log_a b = 1. Valid when a, b ≠ 1.
Chain rule: log_a b × log_b c = log_a c; The middle term b cancels, like fractions. The base of the second log must match the number in the first.
Power of base: log_(b^n) a = (1 ÷ n) × log_b a; Also log_b (a^m) = m × log_b a.
Base-power identity: b^(log_b a) = a; Useful for quick simplification.
Link between ln and log: ln x = ln 10 × log x ≈ 2.3026 × log x; Use only when the value of ln 10 is given or standard.

How to solve Common and Natural Logarithms and Change of Base questions

Use this method for any mixed-base logarithm question.

1Check the conditions: the number must be positive and the base must be positive and not 1.
2Note every base in the question. If they differ, decide on one common base.
3Apply the change of base formula, or the reciprocal or chain rule, to bring all terms to that base.
4Use the laws of logarithms (product, quotient, power) to combine or split terms.
5If the unknown is in the argument, convert to exponential form: log_b a = x gives a = b^x.
6If the unknown is in the base or exponent, express both sides with the same base or take logs of both sides.
7Substitute the given values such as log 2 = 0.3010 and compute.
8Check that the answer keeps every logarithm's number positive, then match it with the options.

Quickest way: Spot the chain and the reciprocal

When to use it: Use when the question multiplies logs with bases and numbers that link up, or gives log values and asks for a log in another base.

Look for a pattern like log_a b × log_b c. The middle term b cancels, so the answer is log_a c.
If a log has base and number swapped, use the reciprocal rule instead of calculating.
Write bases as powers of a small number, such as 8 = 2³ and 27 = 3³, then pull the power out.
For a value like log_4 5 with log 2 given, rewrite as log 5 ÷ log 4 and use log 5 = 1 − log 2.
Test options by plugging in simple numbers: if x = 8 and the base is 2, the log is 3.
If a question needs a long decimal calculation and the options are close, skip it and return later.

Common mistakes in Common and Natural Logarithms and Change of Base

Writing log_b a = log b ÷ log a (upside down).
Students remember the formula as a fraction but not which log goes on top.
Fix: The number goes on top and the base goes below: log_b a = log a ÷ log b. Check with log_2 8 = log 8 ÷ log 2 = 3.
Treating log (a + b) as log a + log b.
The product law log (ab) = log a + log b is over-applied.
Fix: The product law works only for multiplication. log (a + b) has no simple expansion.
Assuming log with no base is always base e.
Calculus and calculators use ln or log for base e in some books.
Fix: In CA Foundation, log usually means base 10 and ln means base e. Read the question for any stated base.
Writing log (a ÷ b) as log a ÷ log b.
Mixing the quotient law with the change of base fraction.
Fix: log (a ÷ b) = log a − log b. A fraction of logs, log a ÷ log b, is a change of base to log_b a.
Ignoring the domain and accepting a negative argument.
Solving the equation algebraically and forgetting to check.
Fix: Put each root back into the original logarithms. Reject any root that makes a number zero or negative.
Using the wrong value of ln 10 or confusing 2.3026 with log 10.
Both numbers appear in the same tables and notes.
Fix: log 10 = 1 in base 10. ln 10 ≈ 2.3026 in base e. They are different quantities.

Worked examples

Example 1

The value of log_2 3 × log_3 4 × log_4 5 × log_5 8 is: (a) 2 (b) 3 (c) 4 (d) 6

Show the solution

Use the chain rule: log_a b × log_b c = log_a c. The base of the second log must match the number in the first.
log_2 3 × log_3 4 = log_2 4 (here a = 2, b = 3, c = 4).
log_2 4 × log_4 5 = log_2 5 (here a = 2, b = 4, c = 5).
log_2 5 × log_5 8 = log_2 8 (here a = 2, b = 5, c = 8).
Since 2³ = 8, log_2 8 = 3.

Answer: (b) 3

Example 2

If log 2 = 0.3010, the value of log_4 5 is closest to: (a) 0.699 (b) 1.161 (c) 0.602 (d) 2.322

Show the solution

Change to base 10: log_4 5 = log 5 ÷ log 4.
log 5 = log (10 ÷ 2) = 1 − log 2 = 1 − 0.3010 = 0.6990.
log 4 = 2 × log 2 = 0.6020.
log_4 5 = 0.6990 ÷ 0.6020 ≈ 1.161.

Answer: (b) 1.161

Example 3

If log_x 81 = 4 and log_2 y = 5, then x + y equals: (a) 32 (b) 35 (c) 38 (d) 41

Show the solution

log_x 81 = 4 means x⁴ = 81.
Since 3⁴ = 81 and the base must be positive, x = 3.
log_2 y = 5 means y = 2⁵ = 32.
x + y = 3 + 32 = 35.

Answer: (b) 35

Exam tips

Chain-rule questions are fast marks. Scan for matching middle terms before you calculate anything.
Memorise log 2 = 0.3010, log 3 = 0.4771 and log 5 = 1 − log 2. Most numeric options come from these.
Convert bases to powers (4 = 2², 9 = 3², 8 = 2³) before using the power-of-base rule.
With 0.25 negative marking, skip a heavy decimal question if options are close. Return if time remains.
Eliminate options using size: if log_4 5 must be slightly above 1 because 5 is just above 4, discard values below 1.

Practice questions from Ratio and Proportion, Indices and Logarithms

Common and Natural Logarithms and Change of Base: frequently asked questions

What is the difference between common and natural logarithm?

A common logarithm uses base 10 and is written log. A natural logarithm uses base e, about 2.718, and is written ln. Both follow the same laws, only the base differs.

What is the change of base formula for logarithms?

log_b a = log_c a ÷ log_c b, where c is any positive base other than 1. Put the number on top and the old base below. You can choose c as 10 or e to suit the question.

Is log x in CA Foundation base 10 or base e?

Usually it is base 10, and ln x is base e. If a question states a base, follow it. Read the question carefully before you start.

How do I solve a logarithm equation step by step?

Check the domain, bring all logs to one base, combine them using the laws, convert to exponential form and solve. Then verify that every logarithm's number stays positive.