SOLUTION: Solve the following logarithmic equation: log (base: 5) (5- 4 x) = log (base: √5) (2 - x) *The base of the second logarithm is the square root or under root of 5. :) =)

Question 398840: Solve the following logarithmic equation:
log (base: 5) (5- 4 x) = log (base: √5) (2 - x)
*The base of the second logarithm is the square root or under root of 5. :) =)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Solving equations where the variable is in the argument to a logarithm usually starts with transforming the equation into one of the following forms:
log(expression( = other-expression
or
log(expression) = log(other-expression)

It may appear that your equation is already in the second form. But the logarithms in the second form need to be of the same base. And your equation has logarithms of different bases. So we start by using the base conversion formula, , to convert one of the logarithms into an expression of logarithms of the other's base. I will use this to convert the logarithm on the right into base 5 logarithms:

The denominator on the right represents the exponent for 5 that results in . If you remember the relationship between radicals and fractional exponents you will realize that square roots are just powers of 1/2. So the denominator is 1/2:

Simplifying we get:

The logarithms are of the same base now. But there is that 2 in front on the right side. Fortunately a property of logarithms, , allows us to move a coefficient into the argument as an exponent:

We finally have the second form. The second form tells us that two logarithms of the same base are equal. In order for these logarithms to be equal the arguments must be equal. So:

Now we solve this equation. First we simplify:

This is a quadratic equation (because of the ) so we want one side to be zero. Subtracting 5 and adding 4x we get:

Next we factor (or use the Quadratic Formula). This factors easily as a difference of squares:
0 = (x+1)(x-1)
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:
x+1 = 0 or x-1 = 0
Solving these we get:
x = -1 or x = 1

When solving logarithmic equations like this one, you must check your answers! You must ensure that your solutions make the arguments of the logarithm(s) positive. "Solutions" which make any logarithm's argument zero or negative must be rejected. And these "solutions" can happen even if you have made no mistakes! This is why you must check your answers.

Always use the original equation when checking:

Checking x = -1:

We can see already that both arguments are positive (even though x itself was negative). So there is no reason to reject this solution. This is the required part of the check. The rest of the check will tell us if we made a mistake. You are welcome to finish the check.

Checking x = 1:

We can see already that both arguments are positive. So there is no reason to reject this solution, either.

So there are two solutions to your equation: x = -1 or x = 1.
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SOLUTION: Solve the following logarithmic equation: log (base: 5) (5- 4 x) = log (base: &#8730;5) (2 - x) *The base of the second logarithm is the square root or under root of 5. :) =)

SOLUTION: Solve the following logarithmic equation: log (base: 5) (5- 4 x) = log (base: √5) (2 - x) *The base of the second logarithm is the square root or under root of 5. :) =)