Better Students Ask More Questions.
Solve in x equation log(base x) 3+ log(base `sqrtx` )3 = 15
1 Answer | add yours
You need to use the following logarithmic identity, such that:
`log_ a b = 1/(log_b a)`
Reasoning by analogy yields:
`log_x 3 = 1/(log_3 x)`
`log_(sqrt x) 3 = 1/(log_3 sqrt x) => log_(sqrt x) 3 = 1/(log_3 (x^(1/2)))`
Using the power property of logarithms, yields:
`log_(sqrt x) 3 = 1/((1/2)log_3 x)`
You need to re-write the equation, such that:
`1/(log_3 x) + 2/(log_3 x) = 15`
You should come up with the following substitution, such that:
`log_3 x = t`
Changing the variable in equation, yields:
`1/t + 2/t = 15 => 1 + 2 = 15t => 3 = 15t => t = 3/15 => t = 1/5`
Replacing back the original variable, yields:
`log_3 x = 1/5 => x = 3^(1/5) => x = root(5) 3`
Since the value `x = root(5) 3` is positive, you do not need to test its validity.
Hence, evaluating the solution to the given equation, yields `x = root(5) 3.`
Posted by sciencesolve on August 3, 2013 at 4:21 PM (Answer #1)
Join to answer this question
Join a community of thousands of dedicated teachers and students.