SOLUTION: Solve the following equation A) 9 log 5 base x = log x base 5 B) log x/2 base 8 = log x base 2

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve the following equation A) 9 log 5 base x = log x base 5 B) log x/2 base 8 = log x base 2      Log On


   

Question 1088005: Solve the following equation
A) 9 log 5 base x = log x base 5
B) log x/2 base 8 = log x base 2
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the following equation
A) 9 log 5 base x = log x base 5
B) log x/2 base 8 = log x base 2
—————————————————————————
A) I don't see how to solve this without using an iterative technique
+9log%28x%2C%285%29%29+=+log%285%2C%28x%29%29+
+log%28x%2C%285%29%29+=+%281%2F9%29%2Alog%285%2C%28x%29%29+



— Maybe another tutor has a better solution. I would be tempted to iterate the last equation until it is (close enough to) zero.
—————————————————————————
This one is easily solved
B) +log%288%2C+%28x%2F2%29%29+=+log%282%2C%28x%29%29+



+matrix%281%2C3%2C++x%2F2+%2C+%22=%22%2C+x%5E3+%29+
+matrix%281%2C3%2C++1%2F2+%2C+%22=%22%2C+x%5E2+%29+
+highlight%28+matrix%281%2C3%2C++x+%2C+%22=%22%2C++sqrt%282%29%2F2%29+%29+
——
7/18/17 - I can't believe I overlooked that property, +log%28a%2Cb%29+=+1%2F%28log%28b%2Ca%29%29+, which is easy enough to derive from the change-of-base calculation. Thanks tutor Ikleyn!


Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve the following equation
A) 9 log 5 base x = log x base 5
~~~~~~~~~~~~~~~~~~~~~~~~ 
9%2Alog%285%2C%28x%29%29 = log%28x%2C%285%29%29.   (1)


There is the basic property of logarithms  log%28a%2C%28b+%29%29 = 1%2Flog%28b%2C%28a%29%29.

Apply it, and your equation  (1) becomes

9%2Alog%285%2C%28x%29%29 = 1%2Flog%285%2C%28x%29%29  ====>

%28log%285%2C%28x%29%29%29%5E2 = 1%2F9  ====>

log%285%2C%28x%29%29 = sqrt%281%2F9%29  ====>


====  There are actually TWO solutions:


1)  log(5,(x))}}} = 1%2F3  ====>  x = root%283%2C5%29,     and


2)  log(5,(x))}}} = -1%2F3  ====>  x = 1%2Froot%283%2C5%29.


Answer.  There are TWO solutions:  x = root%283%2C5%29  and  x = 1%2Froot%283%2C5%29.

Solved.


On logarithms and their properties, see the lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Solving logarithmic equations
    - Using logarithms to solve real world problems
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Logarithms".