Question 1088005
.
Solve the following equation
A) 9 log 5 base x = log x base 5
~~~~~~~~~~~~~~~~~~~~~~~~

<pre>
{{{9*log(5,(x))}}} = {{{log(x,(5))}}}.   (1)


There is the basic property of logarithms  {{{log(a,(b ))}}} = {{{1/log(b,(a))}}}.

Apply it, and your equation  (1) becomes

{{{9*log(5,(x))}}} = {{{1/log(5,(x))}}}  ====>

{{{(log(5,(x)))^2}}} = {{{1/9}}}  ====>

{{{log(5,(x))}}} = {{{sqrt(1/9)}}}  ====>


====  There are actually TWO solutions:


1)  log(5,(x))}}} = {{{1/3}}}  ====>  x = {{{root(3,5)}}},     and


2)  log(5,(x))}}} = {{{-1/3}}}  ====>  x = {{{1/root(3,5)}}}.


<U>Answer</U>.  There are TWO solutions:  x = {{{root(3,5)}}}  and  x = {{{1/root(3,5)}}}.
</pre>

Solved.



On logarithms and their properties, see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/Properties-of-the-logarithm.lesson>Properties of the logarithm</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/change-of-base-formula-for-logarithms.lesson>Change of Base Formula for logarithms</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/How-to-solve-logarithmic-equations.lesson>Solving logarithmic equations</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson>Using logarithms to solve real world problems</A> 

in this site.


Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Logarithms</U>".