<PRE><B>Question 277569</B>
{{{2log((x)) - log((5)) = 4}}}
We want the equation in the form:
log(expression) = other-expression
So somehow we need to combine the two logarithms into one. These two logarithms are not like terms so we cannot subtract them. But there is a property of logarithms, {{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}, which can be used to combine two logarithms if al of the following are true:&lt;ul&gt;&lt;li&gt;there is a &quot;-&quot; between them&lt;/li&gt;&lt;li&gt;the bases of the logarithms are the same&lt;/li&gt;&lt;li&gt;the coefficients of the logarithms are 1&#39;s&lt;/li&gt;&lt;/ul&gt;
Your logarithms meet the first two but not the last. So now our goal is to get rid of the 2 in front of the first log. And fortunately there is another property of logarithms, {{{q*log(a, (p)) = log(a, (p^q))}}}, which can be used to move a coefficient of a logarithm into its argument as an exponent. Using this on your first log we get:
{{{log((x^2)) - log((5)) = 4}}}
These are still not like terms so we still cannot subtract them. But we can now use the other property to combine them:
{{{log((x^2/5)) = 4}}}
We now have the desired form. Once we have this form the next step is to rewrite the equation in exponential form:
{{{x^2/5 = 10^4}}}
which simplifies to:
{{{x^2/5 = 10000}}}
Now the variable is out of the argument where we can &quot;get at it&quot;. Solving this for x we start by multiplying both sides by 5 to get rid of the fraction:
{{{x^2 = 50000}}}
So
{{{x = sqrt(50000)}}} or {{{x = -sqrt(50000)}}}
Simplifying these square roots we get:
{{{x = 100sqrt(5)}}} or {{{x = -100sqrt(5)}}}&lt;br&gt;
When solving logarithmic equations, it is important (not just a good idea) to check your answers. Even if we&#39;ve done everything correct so far, we need to make sure that each answer makes the argument of any logarithms positive.&lt;br&gt;
Always use the original equation to check:
{{{2log((x)) - log((5)) = 4}}}
Checking {{{x = 100sqrt(5)}}}
{{{2log((100sqrt(5))) - log((5)) = 4}}}
Both arguments are positive so it looks good. (You&#39;re welcome to finish the check on your own.)&lt;br&gt;
Checking {{{x = -100sqrt(5)}}}
{{{2log((-100sqrt(5))) - log((5)) = 4}}}
As you can see the argument to the first logarithm is negative. We cannot allow this to occur. So we reject this solution.&lt;br&gt;
The only solution to your equation, then, is {{{x = 100sqrt(5)}}}</PRE>