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# Solving Equations with Log Terms on Each Side

Example 1

Solve: 2ln(x - 4) = ln(40 - 6x)

 Solution Use the Log of a Power Property. Use the Principal of Logarithmic Equality. Square the left side. Add 6x to both sides. Subtract 40 from both sides. Factor. 2ln(x - 4) ln(x - 4)2 (x - 4)2 x2 - 8x + 16 x2 - 2x + 16 x2 - 2x - 24 (x + 4)(x - 6) = ln(40 - 6x) = ln(40 - 6x) = 40 - 6x = 40 - 6x = 40 = 0 = 0 Use the Zero Product Property. Solve for x. x + 4 x = 0 = -4 or or x - 6 = 0 x = 6

It appears that there are two solutions. However, we can only take the log of a positive number. Therefore, we must check each answer to be sure that it leads to taking the log of a positive number.

 Check the first solution, x = -4: Substitute -4 for x. Simplify.Since ln(-8) is undefined, x = -4 is not a solution. 2ln(x - 4) 2ln(-4 - 4) 2ln(-8) = ln(40 - 6x) = ln(40 - 6(-4)) ? = ln(64) ? Check the second solution, x = 6: Substitute 6 for x. Simplify. Use the Log of a Power Property. Simplify. Is Is Is Is 2ln(x - 4)2ln(6 - 4) 2ln 2 ln 22 ln 4 = ln(40 - 6x) = ln(40 - 6(6)) ? = ln 4 ? = ln 4 ? = ln 4 ? Yes
So, x = 6 is the solution.

Example 2

Solve: 2log x = log 144

 Solution Use the Log of a Power Property. Use the Principal of Logarithmic Equality. Take the square root of both sides. Simplify. 2log x log x2 x2 x x = log 144= log 144 = 144 = Â± = Â± 12
The only solution is x = - 12 since log(-12) is undefined.

We leave the check to you.

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