SOLUTION: ln(x) + ln(x � 5) = 4 Solve the logarithmic equation algebraically Approximate the result to three decimal places

Question 628984: ln(x) + ln(x � 5) = 4
Solve the logarithmic equation algebraically
Approximate the result to three decimal places
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
ln(x) + ln(x � 5) = 4
Solving equations where the variable is in the argument of one or more logarithms usually starts with using algebra and/or properties of logarithms to transform the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)

Since your equation has the "non-log" term of 4, it will be more difficult to transform it into the "all-log" second form.

So we will aim for the first form. The first form has a single logarithm. Your equation has two. Somehow we need to combine the two logs into one (or find some other way to make one of the logs disappear). There are two ways to combine logs:

Add or subtract them if they are like terms. Like logarithmic terms have the same bases and the same arguments.
Use one of the following properties of logarithms:

Your logs have the same base, e. but they have different arguments, x and x-5, so we will not be able to add them together.

Your logs have the same base and their coefficients are 1's so we will be able to use the properties to combine them. We use the first property because its logs, like ours, have a "+" between them:
ln(x*(x � 5)) = 4
which simplifies to:

We now have the desired form.

The next step with the first form is to rewrite the equation in exponential form. In general, is equivalent to . Using this pattern on our equation we get:

This is an equation we can solve. It is a quadratic equation so we want one side to be zero. Subtracting from each side we get:

With the this will be difficult to factor. So we will use the Quadratic Formula:

Simplifying:

which is short for:
or
These are exact expressions for possible solutions to your equation. (I'll explain shortly why I described the solutions as "possible".) The problem asks for decimal approximations of the solution(s). This is as good a time as any for this. So we'll get out our calculators. (If your calculator does not have a button for the number "e", then find the inverse ln of 1. If you can't figure that out, type in 2.71828183 in place of e.) Using our calculators on the above and rounding the answer down to 3 decimal places we should get:
x = 10.301 or x = -5.301

When solving logarithmic equations like these we must check our solutions. It is not optional! You must ensure that all arguments to all logs remain positive for each solution you found. If you find that a "solution" makes an argument of a log zero or negative, then you must reject/discard that solution. (This is why I used "possible" to describe the solutions we have so far.)

Use the original equation to check:
ln(x) + ln(x � 5) = 4

Checking x = 10.301:
ln(10.301) + ln(10.301 � 5) = 4
We can already see that both arguments will be positive. So this solution checks.

Checking x = -5.301:
ln(-5.301) + ln(-5.301 � 5) = 4
We can already see that both arguments will be negative. So this solution fails the check. SO we reject this solution. (Note: If only one argument had been zero or negative we would still reject the solution! Arguments to logs can never be zero or negative!)

So there is only one solution to your equation: approximately 10.301
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