Solving log functions
1) log x + log(x-6) = log 7
adding logs is same as multiplying the arguements.
LogA+LogB= log(AB)
log(x(x-6))=log7
take the antilog of each side.
x(x-6)=7
solve.
Note that
x^2 -6x -7 = 0
is easily factored.
There are two solutions that are integers. Only the positive root is allowed, however, since the log of a negative number is undefined.
To solve the given log equation, we need to use logarithmic properties and simplify the equation to a form where we can isolate the variable, x.
Let's start by applying the logarithmic property which states that log a + log b = log (a * b). Using this property, we can rewrite the equation as:
log(x) + log(x-6) = log(7)
log(x * (x-6)) = log(7)
Now, using the property that log a = log b is equivalent to a = b, we can remove the logarithms from both sides of the equation:
x * (x-6) = 7
Now, we have a quadratic equation. Let's solve it by rearranging the equation and setting it equal to zero:
x^2 - 6x - 7 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Let's factor it:
(x - 7)(x + 1) = 0
This gives us two possible solutions for x:
x - 7 = 0 or x + 1 = 0
Solving for x in each equation:
x = 7 or x = -1
So, the solutions to the given log equation are x = 7 and x = -1.