Exponential & Log Functions

Question

If⁡ log2𝑛 (1944)= log𝑛 (486√2), find the values of 𝒏𝟑

Answer 1

Use the change of base formula to convert both sides of the given equation to base 10 logs:

 

We get (log 1944)/(log(2n) = (log (486√2))/(logn).

 

So, (log 1944)(logn) = (log2 + logn)(log(486√2))

 

(log1944)(logn) = (log2)(log(486√2)) + (logn)(log(486√2))

 

(logn)(log1944 - log(486√2)) = (log2)(log(486√2)

 

logn = (log2)(log(486√2))/(log(1944) - log(486√2))

 

logn = 1.891434178, so n = 10^1.891434178

 

Therefore, n³ = 10^5.67430251 

Answer 2

Let's set A= log_2n(1944)=log_n(486 √2), then we have:

1. (2n)^A=1944
2. n^A=486√2

Now by dividing the above equations we obtain:

1. (2n/n)^A=1944/(486√2) which is 2^A=4/√2 or 2^A= (2)^3/2 implying that A=3/2.

Plugging the value of A in the second equation above: n^(3/2)=486√2 or √n³ = √2.√486². Subsequently, we obtain: n³ = 2*(486)²= 472392.