# SOLVING LOGARITHMIC EQUATIONS

1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

Example 6: Solve for x in the equation

Solution:

Step 1: The term Ln(x-5) is valid when x>5, the term Ln(10-x) is valid when x<10, the term Ln(x-6) is valid when x>6, and the term Ln(x-1) is valid when x>1. If we restrict the domain to the set of all real numbers x between 6 and 10 or 6<x<10, every term in the equation is valid.
Graph the function

[formed by subtracting right side of original problem from the left side of equation] and note that the entire graph is located between the values of x = 6 and x = 10.
Note also that the graph crosses the x-axis at 7. This means that the solution to the problem is 7.

Step 2: Simplify both sides of the original equation by combining the logarithmic terms according to Logarithmic Rule 1:

Step 3: The above equation has the same form as the equation Ln(a) = Ln(b). The a must equal the b for the equation to be valid. Therefore,

Step 4: Expand each side of the above equation:

Step 5: Add to both sides:

Step 6: Subtract 15x from both sides of the above equation:

Step 7: Add 50 to both sides of the above equation:

Step 8: Divide both sides of the above equation by 2:

Step 9: Solve for x using the quadratic formula:

x=7,4. Only 7 is in the interval (6,10).

If you would like to review another example, click on Example.

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