CAN U CRACK 9^x 27^x = 36 WITH LAWS OF INDICES?

EXPONENTIAL EQUATIONS

🔹 Prime Factorization & Base Conversion

Always try to rewrite numbers as powers of a common prime base (e.g., 2,3,52, 3, 5).

This helps simplify expressions and compare exponents directly.

Key Concept:

Prime Factorization Principle – expressing numbers as products of prime powers.

🔹 Laws of Indices (Exponent Rules)

Power of a Power: (am)n=amn(a^m)^n = a^{mn}

Product of Same Base: am⋅an=am+na^m \cdot a^n = a^{m+n}

Quotient of Same Base: aman=am−n\dfrac{a^m}{a^n} = a^{m-n}

Zero Exponent Law: a0=1a^0 = 1 (for a≠0a \neq 0)

Negative Exponent: a−n=1ana^{-n} = \dfrac{1}{a^n}

Terminology: Indices / Exponents represent repeated multiplication.

🔹 Exponential Equations

An equation where the unknown variable appears in the exponent.

Solving usually involves:

Rewriting in a common base, then comparing exponents.

Using logarithms if exact base matching isn’t possible.

🔹 Logarithms (Inverse of Exponentials)

Definition: If ab=ca^b = c, then log⁡ac=b\log_a c = b.

Used to “bring down” exponents and turn an exponential equation into a linear one.

Key Properties of Logarithms:

log⁡(ab)=log⁡a+log⁡b\log(ab) = \log a + \log b

log⁡(ab)=log⁡a−log⁡b\log\left(\dfrac{a}{b}\right) = \log a - \log b

log⁡(an)=nlog⁡a\log(a^n) = n \log a

Change of Base Formula:

log⁡ab=ln⁡bln⁡a=log⁡10blog⁡10a\log_a b = \frac{\ln b}{\ln a} = \frac{\log_{10} b}{\log_{10} a}

🔹 Related Concepts to Master

Exponential Growth & Decay

Graphs of Exponential & Logarithmic Functions

Comparison of Exponential Equations (when bases are the same, equate powers).

Application in Olympiad/Competitive Math – many problems test prime factorization + indices rules.

exponential equations, laws of indices, power rule, product rule, logarithm, change of base, algebraic manipulation, Olympiad mathematics, solving with logs, exponent rules.

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