Solving Log Functions:

STEP 1: Identify the type of Log function.

a: Single logarithmic term

b. Multiple logarithmic terms

c. Logarithmic term and exponential term

STEP 3: Solve

To the greatest extent possible, simplify the equation using the rules of logarithms.

STEP 2: Set equations to equal each other.

Take the logarithm of both sides of the equation using the same base. This will transform the equation from a logarithmic equation into an algebraic equation.

STEP 4: Check the Solutions:

Substitute the solution back into the original equation to ensure that it satisfies the equation.

EXAMPLE: Step 1: Combine the logarithmic terms

Using the logarithmic property log(a) + log(b) = log(a * b), we can combine the two logarithmic terms on the left side of the equation:

log((2x - 5) * (x - 3)) = 1

Step 2: Rewrite the equation in exponential form

The exponential form of a logarithmic function is base^log= argument. We can rewrite the equation as:

10^1 = (2x - 5) * (x - 3)

Step 3: Simplify the right side of the equation

We can simplify the right side of the equation by multiplying the two binomials:

10 = 2x^2 - 11x + 15

Step 4: Move all terms to one side of the equation

To solve for x, we neeed to get all the terms on one side of the equation. Let's move the constant term to the left side of the equation:

2x^2 - 11x + 5 = 0

Step 5: Factor the quadratic equation

We can factor the quadratic equation to solve for x. We need to find two numbers whose product is 2 * 5 = 10 and whose sum is -11. Those teo numbers are -1 and -10. Therefore:

(2x - 1)(x - 5) = 0

Step 6: Solve for x

We can solve for x by setting each factor equal to zero and solving for x:

2x - 1 = 0 or x - 5 = 0

2x = 1 or x = 5

x = 1/2 or x = 5

Step 7: Check the solution

We need to check if the solution we obtained satisfies the original equation. Let's plug each solution into the original equation:

For x = 1/2:

log(2(1/2)-5)+log(1/2-3) = log(-4) + log(-5/2)

The log function is undefined for negative arguments, so x = 1/2 is not a valid solution.

For x = 5:

log(2(5)-5)+log(5-3) = log(5) + log(2) = log(10)

10^1 = 10

Therefore, x = 5 is the solution to the equation.

Final Answer: x = 5