Answer:
a = 64
b = 2^(3/2)
c = √2
Step-by-step explanation:
Given:
[tex] log_{b}(a) = 4 \\ log_{c}(b) = 3[/tex]
[tex]a = 32c {}^{2} [/tex]
From the above logarithm equations,it is evident that
- b^4 = a ...(i)
- c^3 = b ...(ii)
We work out the value of a,b,c:
Replace the value of a with b⁴ to the last non- logarithmic equation:
Now replace b as c³:
- (b)⁴ = 32c²
- b*b*b*b = 32c²
- c³ * c³ * c³ *c³ = 32c²
Now solve:
Divide both sides by c²:
We know (a^n)/a^b) = a^(n-b)
- c¹²-² = 32
- c¹⁰ = 2⁵ (2⁵ = 32)
Multiply powers by 1/10:
- c¹⁰*¹/¹⁰ = 2⁵*¹/¹⁰
- c = 2^1/2 = √2
Thus, we individually work out a and b:
From equation (ii):
- c³ = b
- (√2)³ = b
- 2^((1/2)*(3)) = b = 2^(3/2)
From equation (i):
- b⁴ = a
- (2³/²)⁴ = a
- a = 2³*² = 2⁶ = 64
