Question

find the value of x if log of x 729 =2​

Answer 1

Answer:

(1.) log₉ √729 = x, but √729 = 27; therefore, we have:

(2.) log₉ 27 = x

The logarithmic equation (2.) is equivalent to following exponential equation (3.):

(3.) 9ˣ = 27

(4.) (3²)ˣ = 3³

By the “Power of a Power” property of exponents, i.e., (bᵐ)ⁿ = bᵐⁿ, where b is a positive real number and m and n are any real numbers¹; therefore, equation (4.) becomes:

(5.) 3²ˣ = 3³

By another property of exponents: “If b > 0, b ≠ 1, and m and n are real numbers, then bⁿ = bᵐ if and only if n = m,”² and we have:

(6.) 2x = 3

Now, solving for x:

(7.) 2x/2 = 3/2

(8.) (2/2)x = 3/2

(9.) (1)x = 3/2

(10.) x = 3/2

Check:

9ˣ = 27

9^(3/2) = 27

(9^½)³ = 27

(3)³ = 27

3³ = 27

27 = 27