Evaluate the expression without a calculator

Question

A) log_49 (7)

B)log_9 (√3)

 

Answer 1

I am assuming the number after the underscore is the base.

 

a)    49^(?) = 7

 

       (7^2)^x = 7^1  <----  49 = 7 * 7 = 7^2

 

       7^(2x) = 7^1  <--- property of exponents; raising a monomial to a power, multiplies the exponents

 

       2x = 1 <--- got 'em with the same base, boss !!! exponents MUST be the same; sets the equal to each other

 

        x= 1/2.  Makes sense.... square root of 49 is 7

 

(B)    9^X = sqrt(3) = 3^(1/2) <--- definition of square root; sqrt(x) = x^(1/2); same as exponent 1/2

 

       (3^2)^X = 3^(1/2)  <----   9 = 3 x 3 = 3^2

 

        3^(2x)  = 3^(1/2) <--- property of exponents; monomial to a power multiplies the exponents

 

     2x = 1/2  <--- got 'em with the same base, so the exponents MUST be the same; sets the equal

 

   x = 1/4

 

  So the square root of the square root of 9 is square root of 3