Logarithmic Series: loge(1+x) Expansion, Formula & Solved Examples (Class 11 Maths)

Logarithmic Series: loge(1+x) Expansion, Formula & Solved Examples (Class 11 Maths)

Appendix 1 of the NCERT Class 11 Maths textbook closes with the logarithmic series, a companion result to the exponential series covered in the previous section. Where the exponential series expresses eˣ as an infinite sum, the logarithmic series does the reverse: it expresses the natural logarithm logₑ(1 + x) as an infinite sum in x. This post states the formula, explains its validity, derives the well-known result for logₑ2, and works through a complete NCERT example involving the roots of a quadratic equation.

The Logarithmic Series Formula

NCERT states the following theorem without proof:

Theorem. If |x| < 1, then logₑ(1 + x) = x - x²/2 + x³/3 - x⁴/4 + ...

The expression on the right-hand side, x - x²/2 + x³/3 - x⁴/4 + ..., is called the logarithmic series. Notice its structure: the signs alternate, and the denominator of the n-th term is simply n, with no factorial involved — a clear contrast with the exponential series eˣ = 1 + x/1! + x²/2! + ..., where factorials grow the denominators very quickly. This difference in structure is exactly what makes the logarithmic series converge more slowly than the exponential series for the same value of x, even though both require |x| < 1 (or, in the logarithmic series' case, a boundary extension up to x = 1 as shown below).

The Special Case: Deriving logₑ2

One of the most important consequences of this theorem is a well-known formula for logₑ2. NCERT notes that although the theorem as stated requires |x| < 1, the expansion of logₑ(1 + x) is actually valid for the boundary value x = 1 as well. Substituting x = 1 directly into the logarithmic series:

logₑ2 = 1 - 1/2 + 1/3 - 1/4 + ...

This is a striking result: the natural logarithm of 2, an irrational number connected to exponential growth, is expressed as the alternating sum of the reciprocals of the positive integers. This series is sometimes called the alternating harmonic series, and its connection to logₑ2 is one of the most quoted identities to come out of the logarithmic series.

Worked Example: Roots of a Quadratic Equation

The featured NCERT example demonstrates how the logarithmic series interacts with algebra involving the roots of a quadratic equation, tying together two separate topics from the Class 11 syllabus.

Problem. If α and β are the roots of the equation x² - px + q = 0, prove that:

logₑ(1 + px + qx²) = (α + β)x - [(α² + β²)/2]x² + [(α³ + β³)/3]x³ - ...

Solution. Since α and β satisfy the given quadratic, two standard relationships hold: the sum of the roots α + β = p, and the product of the roots αβ = q. These follow directly from comparing x² - px + q = 0 with the general form x² - (sum of roots)x + (product of roots) = 0.

Start from the right-hand side of the identity to be proved, and split it into two separate logarithmic series:

RHS = [αx - α²x²/2 + α³x³/3 - ...] + [βx - β²x²/2 + β³x³/3 - ...]

Each bracket is exactly the logarithmic series for logₑ(1 + αx) and logₑ(1 + βx) respectively (replacing x in the general formula with αx and βx). So:

RHS = logₑ(1 + αx) + logₑ(1 + βx)

Using the standard logarithm property that the sum of two logarithms equals the logarithm of the product:

RHS = logₑ[(1 + αx)(1 + βx)] = logₑ[1 + (α + β)x + αβx²]

Now substitute the two relationships established earlier — α + β = p and αβ = q:

RHS = logₑ(1 + px + qx²) = LHS

This completes the proof. The solution assumes both |αx| < 1 and |βx| < 1, which is necessary for each individual logarithmic series to be valid.

Why This Proof Technique Matters

This worked example illustrates a technique that recurs throughout advanced problems on the logarithmic series: rather than expanding logₑ(1 + px + qx²) directly (which would be difficult, since it is not in the simple form logₑ(1 + x)), the expression is factored into two simpler logarithmic pieces, each of which matches the standard formula exactly. This "factor and split" approach — splitting a complicated logarithm of a product into a sum of simpler logarithms, then applying the series formula to each piece separately — is a powerful problem-solving strategy that appears repeatedly once the basic logarithmic series formula is understood.

Logarithmic Series vs Exponential Series: A Quick Comparison

Together, these two series complete Appendix 1's tour of infinite series: starting from the general idea of an infinite sum, through the binomial series for any index, the infinite geometric series, and finally the exponential and logarithmic series that underpin much of calculus.

Keep Learning: Related Reading

• Exponential Series & Euler's Number e: Formula, Derivation & Examples (Class 11 Maths)
• Infinite Geometric Series: Sum to Infinity Formula & Solved Examples (Class 11 Maths)
• Binomial Theorem for Any Index: Formula, Conditions & Special Cases (Class 11 Maths)

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