You can put this solution on YOUR website! log(x+3/10)+log(x)+1=0
Solving equations where the variable is in the argument 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)
With the "non-log" terms of 1 and 0, the second form will be more difficult to achieve. So we will aim for the first form. We want the "non-log" terms on one side and the log terms on the other so we will subtract 1 from each side:
log(x+3)+log(x) = -1
Now we need to find a way to combine the two logs into one. They are not like terms so we cannot just add them together. (Like logarithmic terms have the same bases and the same arguments. Your logarithms have the same base, 10, but different arguments, x+3/10 and x.
Fortunately there are two properties of logarithms which allow us to combine two logarithms into one:
These properties require:
Logarithms of the same base.
Coefficients of 1 in front of the two logarithms.
Your logarithms meet both of these requirements. Since your logarithms have a "+" between them, we willuse the first property:
log((x+3/10)*x) = -1
which simplifies to:
Now that we have the first form, the next step is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern on your equation we get: (since the implied base of log is 10)
which simplifies to: (Remember how negative exponents work?)
Now we solve this equation. To make this easier I am going to eliminate the fractions by multiplying by 10:
This is a quadratic equation so we want one side to be zero. Subtracting 1 from each side we get:
Now we factor (or use the Quadratic Formula). This does factor:
5x-1)(2x+1) = 0
From the Zero Produce property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
5x-1 = 0 or 2x+1 = 0
Solving these we get:
x = 1/5 or x = -1/2
Solutions for equations like this must be checked. We must ensure that solutions make the arguments of all logarithms positive. If a "solution" makes an argument zero or negative we must reject that solution! And these rejected "solutions" can occur even if no mistakes were made! So even expert Mathemeticians must check.
When checking use the original equation:
log(x+3/10)+log(x)+1=0
Checking x = 1/5:
log((1/5)+3/10)+log(1/5)+1=0
We can see already that the arguments will both be positive. (We haven't figured out 1/5 + 3/10 but we should know that whatever it is it will be positive.) So there is no reason to reject this solution. This is the required part of the check. The remainder of the check is optinoal and will tell us if we made a mistake. You are welcome to finish it.
Checking x = -1/2:
log((-1/2)+3/10)+log(-1/2)+1=0
We can see already that the second argument is negative. (We haven't figured out -1/2 + 3/10 yet but it doesn't matter how it works out since the other argument is negative.) So we must reject this "solution".
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