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Exponential and Logarithmic Equations

Exponential and Logarithmic Equations. Objectives. Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations. Vocabulary. exponential equation logarithmic equation.

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Exponential and Logarithmic Equations

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  1. Exponential and Logarithmic Equations Objectives Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations.

  2. Vocabulary exponential equation logarithmic equation

  3. An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations: Try writing them so that the bases are the same Take the logarithm of both sides

  4. Solving Exponential Equations by Expressing Each Side as a Power of the Same Base Express each side as a power of the same base. Set the exponents equal to each other. Rewrite the equation in the form Set Solve for the variable.

  5. Solving Exponential Equations Solve and check. 98 – x = 27x – 3 Rewrite each side with the same base; 9 and 27 are powers of 3. (32)8 – x = (33)x – 3 To raise a power to a power, multiply exponents. 316 – 2x = 33x – 9 Bases are the same, so the exponents must be equal. 16 – 2x = 3x – 9 x = 5 Solve for x.

  6. Check 98 – x = 27x – 3 98 – 5275 – 3 93272 729 729  The solution is x = 5.

  7. Solve and check: Rewrite each side with the same base. Now that the bases are the same, solve for

  8. Check: Substitute 4 for the variable.

  9. Solve and check: Rewrite each side with the same base. Now that the bases are the same, solve for

  10. Solve and check: Substitute 3 for the variable and solve.

  11. log5 log5 x–1 = log4 log4 x = 1 + ≈ 2.161 Solving Exponential Equations Solve and check. 4x – 1= 5 We cannot get common bases, so take the log of both sides. log 4x – 1 =log5 Apply the Power Property of Logarithms. (x– 1)log 4 = log 5 Divide both sides by log 4. CheckUse a calculator. The solution is x ≈ 2.161.

  12. Using Logarithms to Solve Exponential Equations Isolate the exponential expression. Take the natural logarithm on both sides of the equation for bases other than 10. Take the common logarithm on both sides of the equation for base 10. Simplify using one of the following properties: or or Solve for the equation.

  13. Solve: Take the natural logarithm on both sides Use the power rule = Solve for x by dividing both sides by Use calculator. • Check.

  14. Solve: Take the natural logarithm of both sides. When you take the natural logarithm of base e, the ln e drops from the equation leaving only the exponent as seen above. This is using the inverse property Also, Check your answer.

  15. Solve: Add 3 to both sides Divide both sides by 40 Take the natural logarithm of both sides Use the inverse property 2.99 Divide both sides by 0.6 and solve for x

  16. Solve Take the natural logarithm on both sides Use the power rule Use the distributive property Rearrange terms Factor out x The solution is approximately

  17. Solve: Let Substitute for Factor on the left Set each factor equal to 0 Solve for Replace for Take the natural logarithm of both sides

  18. Using the Definition of a Logarithm to Solve Logarithmic Equations Express the equation in the form Use the definition of a logarithm to rewrite the equation in exponential form. means Solve for the variable. Check your solutions for in the original equation.

  19. Solve and check: Rewrite in exponential form

  20. Solve and check: Divide both sides by 3 Rewrite the natural logarithm showing base e Rewrite in exponential form Divide both sides by 2

  21. Solve and check: Use the product rule Rewrite in exponential form Simplify Set up as a quadratic equation Factor  Always check your answers with original equation.

  22. Using the One-to-One Property of Logarithms to Solve Logarithmic Equations Express the equation in the form The coefficient must be equal to 1 on both sides. Use the one-to-one property to rewrite the equation without the logarithm. If , then . Solve for the variable. Check proposed solutions in the original equation. must be positive.

  23. Solve: () ln Use the quotient rule Use the one-to-one property Cross multiply Use distributive property Set up as a quadratic equation

  24. Factor Set each factor equal to 0 Check by substituting each solution into the original equation.

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