You can put this solution on YOUR website!
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)
With the non-log term of 1 in your equation, the first form will be easier to reach. So we want to find a way to combine the two logarithms into one.
The two logarithms are not like terms so we cannot just add them together. (Like logarithmic terms have logarithms of the same base with the same arguments.)
Fortunately there is a property of logarithms, , which is another way to combine two logarithms which have a "+" between them. This property only requires that the bases be the same and the coefficients of the logarithms are 1's. Your logarithms meet both requirements so we can use it to combine the two logarithms into one:
which simplifies to:
We now have the first form. The next step with this form is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern on your equation we get:
which simplifies to:
This is an equation we can solve. It is quadratic so we want one side to be zero. Subtracting 5 from each side we get:
Next we factor (or use the Quadratic Formula). This factors fairly easily:
(4x-5)(x+1) = 0
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
4x-5 = 0 or x+1 = 0
Solving these we get:
x = 5/4 or x = -1
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.
Always use the original equation to check:
Checking x = 5/4:
which simplifies as follows:
We can see that both arguments (and bases) are positive. So there is no reason to reject this solution. This is the required part of the check. The remainder of the check will tell us if we made a mistake. You are welcome to finish the check.
Checking x = -1:
We can already see that one argument is negative. Arguments of logarithms can never be negative (because it is impossible to raise 5 to any power and get a negative number). So we must reject this solution. (Important: It is not because x was negative that we rejected x = -1 as a solution. We rejected the solution because an argument of a logarithm is negative when x is -1. Sometimes negative values for x are OK. Sometimes positive values for x are rejected. It all depends on how the arguments work out.)
So the only solution for your equation is:
x = 5/4
(