```Question 383875
log(x+3) = 2 + log(x)
To solve equations where the variable is in the argument of a logarithm you often start by transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)<br>
With the "non-log" term of 2, it will be more difficult to transform your equation into the "all-log" second form. So we will try for the first form.<br>
With the first form we want one side of the equation to have no logarithms. So we will subtract log(x) from each side:
log(x+3) - log(x) = 2
Next we want to combine the two logarithms into one somehow. Fortunately there is a property of logarithms, {{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}, which allows us to to just that! We have two logarithms of the same base, with coefficients of 1 and with a minus between them. This fits the pattern of this property so we can use this property to combine the two logarithms into one:
{{{log(((x+3)/x)) = 2}}}
We now have the first form. With the form the next step is to rewrite the equation in exponential form. In general {{{log(a, (p)) = q}}} is equivalent to {{{p = a^q}}}. Using this pattern on your equation (and remembering that the base of "log" is 10) we get:
{{{(x+3)/x = 10^2}}}
which simplifies to:
{{{(x+3)/x = 100}}}
This is an equation we can solve. Start be eliminating the fraction (by multiplying both sides by x:
x+3 = 100x
Next we subtract x from each side:
3 = 99x
and last we divide both sides by 99:
{{{3/99 = x}}}
which simplifies/reduces to:
{{{1/33 = x}}}<br>
With logarithmic equations it is important, not just a good idea, to check your answers. You must ensure that the argument of all logarithms are positive. (If you find that a "solution" makes even one argument of a logarithm zero or negative then you must reject that "solution"!)<br>
Always us the original equation to check:
log(x+3) = 2 + log(x)
Checking x = 1/33:
{{{log((1/33+3)) = 2 + log((1/33))}}}
We can already see that both arguments are going to work out to be positive. So we have no reason to reject this solution. The rest of the check will tell us if we made a mistake. You are welcome to finish the check.```