SOLUTION: Please show the step-by-step solution to: Prove [(log7)4][log4)9]= [(log7)9] Textbook hint: start by dividing each side by [(log7)4]..... Thank you!

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please show the step-by-step solution to: Prove [(log7)4][log4)9]= [(log7)9] Textbook hint: start by dividing each side by [(log7)4]..... Thank you!      Log On


   

Question 170951This question is from textbook LOF Advanced Algebra
: Please show the step-by-step solution to:
Prove [(log7)4][log4)9]= [(log7)9]
Textbook hint: start by dividing each side by [(log7)4].....
Thank you! This question is from textbook LOF Advanced Algebra

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Here's one way to do it:

%28log%287%2C%284%29%29%29%28log%284%2C%289%29%29%29=log%287%2C%289%29%29 Start with the given equation. Note: I'm ONLY going to manipulate the left side. The right side is for comparison


By the change of base formula log%28b%2C%28x%29%29=log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29, we can rewrite log%287%2C%284%29%29 into %28log%2810%2C%284%29%29%29%2F%28log%2810%2C%287%29%29%29


Also, we can rewrite log%284%2C%289%29%29 into %28log%2810%2C%289%29%29%29%2F%28log%2810%2C%284%29%29%29


Use the change of base formula to expand the logarithms (see above)


Highlight the common terms.


Cancel out the common terms.


Simplify


%28%28log%2810%2C%287%29%29%29%2F%28log%2810%2C%287%29%29%29%29=log%287%2C%289%29%29 Combine and multiply the fractions.


log%287%2C%289%29%29=log%287%2C%289%29%29 Use the change of base formula (in reverse) to simplify the left side



So this shows us that %28log%287%2C%284%29%29%29%28log%284%2C%289%29%29%29=log%287%2C%289%29%29



Note: this can be generalized to say that


%28log%28x%2C%28y%29%29%29%28log%28y%2C%28z%29%29%29=log%28x%2C%28z%29%29