SOLUTION: I know the answer is 2 but i dont know how to get it can you please help me? log(base 4)X-log(base 4)(x-1)=(1/2)

Question 706808: I know the answer is 2 but i dont know how to get it can you please help me? log(base 4)X-log(base 4)(x-1)=(1/2)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Solving equations like this usually starts with transforming the equation into one of the following general forms:
log(expression) = number
or
log(expression) = log(other-expression)

With the "non-log" term of 1/2, it will be more difficult to reach the "all-log" second form. So we will aim for the first form. We can reach the first form if we can find a way to combine the two logs on the left side.

The two logs are not like terms so we cannot subtract them. (Like logarithmic terms have the same bases and the same arguments. Our logs have the same bases, 4, but the arguments are different.

Fortunately there is another way to combine logs. There are two properties of logs which allow this:

These properties require the same bases and coefficients of 1. Our logs fit both requirements. We will use the second property there is a "-" between the logs, just like our logs:

We have now reached 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 our equation we get:

Since 1/2 as an exponent means square root the left side simplifies:

Now that the variables are out of the logarithms we can use "regular" algebra to solve for x. First we'll eliminate the fraction by multiplying both sides bu x-1:

which simplifies to:

Subtracting 2x from each side we get:

Dividing (or multiplying) by -1 we get:

Last of all we check the solution. This is not optional! This is required when solving these logarithmic equations and when you multiply both sides of the equation by something that might be zero (like x-1). We have done both.

Use the original equation to check:

Checking x = 2:

Simplifying:

As we've already found so the first log is 2. And the log of 1, no matter what the base, is zero (because any number except zero to the zero power is 1). So we can replace the two logs:

which simplifies to:

Check!! So x = 2 is the solution to the equation.
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