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4.5 Exponential and Logarithmic Equations

4.5 Exponential and Logarithmic Equations. Example 1. SOLVING AN EXPONENTIAL EQUATION. Solve 7 x = 12. The properties of exponents given in Section 4.2 cannot be used to solve this equation, so we apply the property of logarithms.

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4.5 Exponential and Logarithmic Equations

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  1. 4.5 Exponential and Logarithmic Equations Example 1 SOLVING AN EXPONENTIAL EQUATION Solve 7x = 12. The properties of exponents given in Section 4.2 cannot be used to solve this equation, so we apply the property of logarithms.

  2. CautionBe careful when evaluating a quotient like .Do not confuse this quotient with which can be written as In 12 – In 7. We cannot change the quotient of two logarithms to a difference of logarithms.

  3. SOLVING AN EXPONENTIAL EQUATION Example 2 Solve 32x– 1 = 4x+2

  4. SOLVING BASE eEXPONENTIAL EQUATIONS Example 3

  5. SOLVING BASE e EXPONENTIAL EQUATIONS Example 3

  6. Property of Logarithms If x > 0, y > 0, a > 0, and a ≠ 1, then x = y if and only if loga x = loga y.

  7. SOLVING A LOGARITHMIC EQUATION Example 4 Solve log(x + 6) – log(x + 2) = log x.

  8. CautionRecall that the domain of y = logax is (0, ). For this reason, it is always necessary to check the solutions of a logarithmic equation.

  9. SOLVING A LOGARITHMIC EQUATION Example 5 Solve log(3x + 2) + log(x – 1 ) = 1

  10. SOLVING A BASE e LOGARITHMIC EQUATION Example 6 Solve In eIn x–In(x – 3) = In 2

  11. Solving Exponential Or Logarithmic Equations • To solve an exponential or logarithmic equation, change the given equation into one of the following forms, where a and b are real numbers, a > 0 and • a ≠ 1, and follow the guidelines. • a(x)= b • Solve by taking logarithms on both sides. • 2. Loga(x) = b • Solve by changing to exponential form ab = (x).

  12. Solving Exponential Or Logarithmic Equations 3. loga(x) = logag(x) The given equation is equivalent to the equation (x) = g(x). Solve algebraically. 4. In a more complicated equation, such as the one in Example 3(b), it may be necessary to first solve for a(x)or loga(x) and then solve the resulting equation using one of the methods given above. 5. Check that the proposed solution is in the domain.

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