Question 277571
{{{log(5, (3x+10)) - 3log(5, (4))=2}}}
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 all of the following are true:<ul><li>there is a "-" between them</li><li>the bases of the logarithms are the same</li><li>the coefficients of the logarithms are 1's</li></ul>
Your logarithms meet the first two but not the last. So now our goal is to get rid of the 3 in front of the second 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 second log we get:
{{{log(5, (3x+10)) - log(5, (4^3))=2}}}
which simplifies to:
{{{log(5, (3x+10)) - log(5, (64))=2}}}
These are still not like terms so we still cannot subtract them. But we can now use the other property to combine them:
{{{log(5, (3x+10))/64))=2}}}
We now have the desired form. Once we have this form the next step is to rewrite the equation in exponential form:
{{{(3x + 10)/64 = 5^2}}}
which simplifies to:
{{{(3x+10)/64 = 25}}}
Now the variable is out of the argument where we can "get at it". Solving this for x we start by multiplying both sides by 64 to get rid of the fraction:
3x + 10 = 1600
Subtracting 10 from each side:
3x = 1590
Dividing by 3 we get:
x = 530<br>
When solving logarithmic equations, it is important (not just a good idea) to check your answers. Even if we've done everything correct so far, we need to make sure that each answer makes the argument of any logarithms positive.<br>
Always use the original equation to check:
{{{log(5, (3x+10)) - 3log(5, (4))=2}}}
Checking x = 530:
{{{log(5, (3(530)+10)) - 3log(5, (4))=2}}}
which simplifies to
{{{log(5, (1600)) - 3log(5, (4))=2}}}
Both arguments are positive so it looks good. (You're welcome to finish the check on your own.)<br>
The solution to your equation, then, is x = 530}}}