Question 678907
{{{5log(7, (2x))=10}}}
Solving equations like this usually starts with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)<br>
To reach the first form all we have to do is divide both sides by 5:
{{{log(7, (2x))=2}}}<br>
The next step with this form is to rewrite the equation in exponential form. In general {{{log(a, (p)) = q}}} is equivalent to {{{a^q = p}}}. Using this pattern on our equation we get:
{{{7^2 = 2x}}}
which simplifies to:
{{{49 = 2x}}}<br>
Now that the logs are gone we can solve the equation. Dividing both sides by 2 we get:
{{{49/2 = x}}}<br>
When solving logarithmic equations like this you must check your solution(s). <i>It is not optional!</i> You must ensure that all arguments of all logs are positive for each solution. If a "solution" makes any argument of any log zero or negative we must reject that "solution".<br>
Use the original equation to check:
{{{5log(7, (2x))=10}}}
Checking x = 49/2:
{{{5log(7, (2(49/2)))=10}}}
We can already see the argument, 2(49/2), will be positive if x = 49/2. So this solution passes the check! (If 49/2 had failed this check then we would reject it. And since it was the only "solution" we found rejecting it would mean that the equation has no solutions.)<br>
So the only solution to your equation is x = 49/2.