Question

Simplify the following:

`log sqrt(343)/log 49`

Answer 1

Step 1: We know that

`(log ⁡a)/(log ⁡b) = log_b a`

So, `log sqrt(343)/log 49 = log_49 sqrt343`

Step 2: Express both numbers as powers of 7:

⇒ 343 = 7³

`sqrt(343) = (7^3)^(1/2)`

`sqrt(343) = 7^(3/2)`

⇒ 49 = 7²

Step 3: Substitute these powers into the logarithm:

`log_49 sqrt(343) = log_(7^2) 7^(3/2)`

Step 4: Using the rule:

`log_(a^m) a^n = n/m`

We get,

`log_(7^2) 7^(3/2) = (3/2)/2`

`log_(7^2) 7^(3/2) = 3/4`