Question

Log a/b-c = Log b/c-a = Log c/a-b prove that a^a.b^b.c^c= 1​

Answer 1

Solution :-

Let

$\boxed{\sf{\dfrac{log \: a}{b - c} = \dfrac{log \: b}{c - a} = \dfrac{log \: c}{a - b} = \bf{k} }}$

Now,

• log a = k (b - c)

⇒ log₁₀ a = k (b - c)

⇒ a = 10ᵏ ⁽ᵇ ⁻ ᶜ⁾

⇒ a = 10ᵏᵇ ⁻ ᵏᶜ

The question is about aᵃ

⇒ aᵃ = 10ᵏᵃᵇ ⁻ ᵏᵃᶜ

Then,

• log b = k (c - a)

⇒ log₁₀ b = k (c - a)

⇒ b = 10ᵏ ⁽ᶜ ⁻ ᵃ⁾

⇒ b = 10ᵏᶜ ⁻ ᵏᵃ

⇒ bᵇ = 10ᵏᵇᶜ ⁻ ᵏᵃᵇ

Also,

• log c = k (a - b)

⇒ log₁₀ c = k (a - b)

⇒ c = 10ᵏ ⁽ᵃ ⁻ ᵇ⁾

⇒ c = 10ᵏᵃ ⁻ ᵏᵇ

⇒ cᶜ = 10ᵏᵃᶜ ⁻ ᵏᵇᶜ

aᵃ × bᵇ × cᶜ

= (10ᵏᵃᵇ ⁻ ᵏᵃᶜ) × (10ᵏᵇᶜ ⁻ ᵏᵃᵇ) × (10ᵏᵃᶜ ⁻ ᵏᵇᶜ)

= 10ᵏᵃᵇ ⁻ ᵏᵃᵇ ⁻ ᵏᵃᶜ ⁺ ᵏᵃᶜ ⁺ ᵏᵇᶜ ⁻ ᵏᵇᶜ

= 10⁰

= 1

★ HENCE PROVED ★

Answer 2

⁂ REFER TO ATTACHMENT ⁂

߷ SOME IMPORTANT FORMULAS ߷

LOGARITHMS

1. loga mn = logm + logn.

2.log m/n

= logm – logn.

3. loga mn

= n logm.

4. logba =

log a

log a

log alog b .

5. logaa = 1.

6. loga1 = 0.

7. logba = 1/logab

8. loga1= 0.

9. log (m +n) ≠ logm +logn.

10. e

10. e logx = x.

11. logaax

x

x = x.

꧁BE BRAINLY꧂

THANKS!