Question

if the logarithm of 324 is a base a is 4,what is the value of a​

Answer 1

Answer:

a = 3√2

Step-by-step explanation:

We are given,

log(a) 324 = 4

Thus, in exponential form,

a⁴ = 324

a = ⁴√324

a = 324^(¼)

Now,

324 = 2 × 2 × 3 × 3 × 3 × 3

324 = 2² × 3⁴

Now, when its is in the 4th power its prime factorization must be in groups of 4, here, 3 is in group of 4 but 2 is not,

Now,

a = (324)^(¼)

a = (2² × 3⁴)^(¼)

According to the exponential laws,

(a × b)^m = (a^m) × (b^m)

So,

a = (2²)^(¼) × (3⁴)^(¼)

Now, according to the exponential law,

(a^m)^n = a^(m × n)

Thus,

a = 2^(2 × (1/4)) × 3^(4 × (1/4))

a = 2^(1/2) × 3^(1)

a = ²√2 × 3¹

a = √2 × 3

a = 3√2

Hence,

a = 3√2

Hope it helped and believing you understood it........All the best