SOLUTION: log square root(x+3)=1

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: log square root(x+3)=1      Log On


   

Question 408798:
log square root(x+3)=1
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28sqrt%28x%2B3%29%29%29+=+1
Solving equations where the variable is in the argument (or base) of a logarithm usually starts with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)

Your equation is already in the first form. The next step with this form is to rewrite the equation in exponential form. In general log%28a%2C%28p%29%29+=+q is equivalent to p+=+a%5Eq. Using this pattern on your equation (and using the fact that the base of "log" is 10) we get:
sqrt%28x%2B3%29+=+10%5E1
which simplifies to:
sqrt%28x%2B3%29+=+10
This is an equation we can solve. Squaring both sides we get:
x+3 = 100
Subtracting 3 we get:
x = 97
With logarithmic equations like yours you must check your solution(s). You must ensure that they make arguments (and bases) of all the logarithms positive. Any "solution" that makes an argument or base zero or negative must be rejected. And these rejected "solutions" can occur even if no mistakes were made while solving! So you must always check these no matter how good at Math you are.

Also, any time you square both sides of an equation (like we did when we eliminated the square root) you must check your solution. So we have two reasons that require that we check.
Always use the original equation to check:
log%28%28sqrt%28x%2B3%29%29%29+=+1
Checking x = 97:
log%28%28sqrt%2897%2B3%29%29%29+=+1
which simplifies as follows:
log%28%28sqrt%28100%29%29%29+=+1
log%28%2810%29%29%29+=+1
The argument of the log is positive. And since the base 10 log of 10 is, by definition, equal to 1, the solution checks.