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Mastering Logarithmic Equations: Techniques and Strategies

This guide provides a comprehensive overview of solving logarithmic equations, utilizing essential properties and identifying when to apply them. You'll learn to simplify and solve equations, check for extraneous solutions, and employ additional mathematical skills like factoring. Examples illustrate the techniques, such as condensing logs and converting to exponential form. Practice problems at the end challenge you to apply what you've learned. Perfect for those wanting to enhance their algebra skills, particularly in logarithmic functions.

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Mastering Logarithmic Equations: Techniques and Strategies

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  1. Solving Lagarithmic Equations

  2. Solving Log Equations • We will be using all of the properties. • You will have to figure out which ones to use and when to use them. • You will have to use other math skills, like factoring, at times. • You will have to check your answer to see if they are extraneous. • Remember, you cannot take the log of a negative number.

  3. Solving Simple Log Equations • Solve the equation exactly. • log4(2x-3) = log4x + log4(x-2) • log4(2x-3) = log4[x(x-2)] • 2x – 3 = x(x – 2) • 2x – 3 = x2 – 2x • x2 – 4x + 3 = 0 • (x – 3)(x – 1) = 0 • x = 3 x = 1 • x = 3 Use Rule 5 to condense the logs into one. When you have one log on each side, you can drop it. (Just like we did with the exponential functions.) Distribute and combine like terms. Factor and solve.

  4. Solving Simple Log Equations • This one looks hard, but it is still not too bad. • log39x – log3(x-8) = 4 • log3() = 4 • = 34 • = 81 • 9x = 81(x – 8) • 9x = 81x – 648 • -72x = -648 • x = 9 Condense the logs using Rule 6. Rewrite in exponential form What is 34? Cross multiply. Solve for x.

  5. Solving Simple Log Equations • Try a couple on your own. • log(x + 8) = log x + log (x + 3) • x = 2 • log3(x + 1) – log3(x – 4) = 3 • x = 4.19

  6. Solving Log Equations • What if they get a little more difficult? • log(6x + 5) – log 3 = log 2 – log x • log = log • = • 6x2 + 5x = 6 • 6x2 + 5x – 6 = 0 • x2+ 5x – 36 = 0 • (x + 9)(x – 4) = 0 • (x + )(x - ) = 0 • x =

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