You can put this solution on YOUR website! Solving equations where the variable is in the argument of one or more logarithms usually involves transforming the equation into one of the following general forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
In order to transform an equation into one of these forms, we can
replace known logarithms with their values. For example can be replaced by a 3 (because ).
use properties of logarithms
The first two properties allow you to combine two logarithms into one if the bases of the logarithms are the same and if the coefficients of the logarithms are the same. The first one is used when the two logarithms have a "+" between them and the second is used when there is a "-" between them.
The third property is used to move a coefficient into the argument as an exponent. This is important because the two general forms above and the two properties above all require coefficients of 1.
Let's see how this works on your equation.:
2log(x)= log(2) + log(x+4)
Since we have nothing but logarithms in this equation and none of these logarithms are known, the first form, whose right side has no logarithms, will be difficult to achieve. So we'll aim for the second form.
On the left side all we need to do is change the coefficient into a 1. This is a job for the third property. Using this property we can move the 2 into the argument as an exponent giving us:
On the right side we have two logarithms of the same base, 10, which have coefficients of 1. We can use the first property (because of the "+" between them) to combine them:
We have now achieved the second general form. With this form the next step is based on some simple logic. If the logarithm of is the same as the logarithm of 2(x+4), then must be the same as 2(x+4):
We now have a quadratic equation to solve. So we want one side to be zero:
Simplifying we get:
Now we can factor (or use the Quadratic Formula). This factors easily:
(x+2)(x-4) = 0
By the Zero Product Property we know that this product is zero only if one of the factors is zero. So:
x+2 = 0 or x-4 = 0
Solving these we get:
x = -2 or x = 4
When solving logarithmic equation it is important, not just a good idea, to check your answers. (Always use the original equation to check.)
2log(x)= log(2) + log(x+4)
Checking x = -2:
2log((-2))= log(2) + log((-2)+4)
which simplifies to:
2log(-2)= log(2) + log(2)
At this point we have a problem. Arguments of logarithms must never be zero or negative. The first logarithm has a negative argument. For this reason we must reject this solution. This is an example of why we need to check solutions to logarithmic equations. We made no mistakes finding x = -2. But it is not a solution to the equation. We can only find these non-solutions by checking.
Checking x = 4:
2log((4))= log(2) + log((4)+4)
which simplifies to:
2log(4)= log(2) + log(8)
The arguments all look good. They are all positive. We can continue the check by using properties:
log(16) = log(16) Check!
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