# Thread: Logarithmic Equation Help

1. ## Logarithmic Equation Help

Howdy friends --

I'm in business calculus at a college in New Jersey and I recently was given this problem to solve on an online homework assignment. The question is to solve for x in terms of k and there is only one line for an answer.

I have already tried submitting x=9^k and x=(9^k)-10, and neither are satisfactory to complete the problem.

Many thanks,
flk

2. Originally Posted by funnylookinkid

Howdy friends --

I'm in business calculus at a college in New Jersey and I recently was given this problem to solve on an online homework assignment. The question is to solve for x in terms of k and there is only one line for an answer.

I have already tried submitting x=9^k and x=(9^k)-10, and neither are satisfactory to complete the problem.

Many thanks,
flk
remember your formulas for summing logs, and taking the inverse of logarithms.

$\displaystyle \log_9 x + \log_9 (x + 10) = k$

$\displaystyle \Rightarrow \log_9 [x(x + 10)] = k$

$\displaystyle \Rightarrow 9^k = x(x + 10)$

this is a quadratic equation, use the quadratic formula to solve

3. Thank you so much! It worked brilliantly. I never would have thought to set the equation up that way!

4. Originally Posted by Jhevon
$\displaystyle \Rightarrow 9^k = x(x + 10)$

this is a quadratic equation, use the quadratic formula to solve
Well from here we can save some of time by just completing the missing square.

$\displaystyle 9^k+25=x^2+10x+25\implies9^k+25=(x+5)^2.$ The rest follows.