You can put this solution on YOUR website! log(base 3) x(x-24) = 4
log(base 3) (x^2-24x)=4
x^2-24x = 3^4
X^2-24x = 81
(x-12)^2 - 144 = 81
x = +/-sqrt(225) +12
x = 15+12 or x = -15+12
x must be positive as can not take a log of a negtive number so
x = 27
You can put this solution on YOUR website!
When you are trying to solve for a variable which is in the argument(s) of logarithms, we will want to:
Transform the equation into one of the following two forms:
log(expression) = another-expression
log(expression) = log(another-expression)
If you have the first form above, rewrite the equation in exponential form. If you have the second form, then just remove the logs. In other words, with the second form, if log(expression) = log(another-expression) then expression = another-expression.
Solve the resulting equation (which will no longer have logarithms).
Since our equation has a term with no logarithms, the 4, we will aim for the first form. This means we will have to combine the two logarithms we have into one. Fortunately we have properties of logarithms that allow us to combine two logs into one:
Since our two logs are added together we will use the first property. (The second has a subtraction between the logs.) So our two logs can combine according to this property into:
And we have achieved the first form.
Now we rewrite the equation in exponential form using the fact that is equivalent to :
The equation is now a quadratic equation. Simplifying both sides we get:
Next we get one side equal to zero by subtracting 81 from each side:
Now we either factor the left side or use the Quadratic Formula. The left side factors fairly easily:
According to the Zero Product Property this product can be zero only if one (or more) of the factors is zero. So: or
Solving these we get: or
When solving logarithmic equations it is more than just a good idea to check your answers. It is important. With logarithms we must make sure their arguments never become negative or zero. And if any values for x makes the argument of a logarithm negative or zero, we must reject those values.
When checking always use the original equation:
Checking x = 27:
So far so good. The arguments of both logarithms are positive. Since and we get: Check!
Checking x = -3:
As we can see, when x = -3 the arguments of both logarithms are negative. So we must reject x = -3. (If x = -3 had made just one argument negative or zero we would still have to reject it.)
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