 Solving Exponential & Logarithmic Equations Properties of Exponential and Logarithmic Equations Let be a positive real number such that , and let and be real numbers. Then the following properties are true: 1. 2. Inverse Properties of Exponents and Logarithms Base a Natural Base e 1. 2. Solving Exponential and Logarithmic Equations 1.To solve an exponential equation, first isolate the exponential expression, then take the logarithm of both sides of the equation and solve for the variable. 2.To solve a logarithmic equation, first isolate the logarithmic expression, then exponentiate both sides of the equation and solve for the variable. For Instance: If you wish to solve the equation, , you exponentiate both sides of the equation to solve it as follows: Original equation Exponentiate both sides Inverse property Or you can simply rewrite the logarithmic equation in exponential form to solve (i.e. ). Note: You should always check your solution in the original equation. Example 1: Solve each equation. a.! " #! b.$% Solution: a.! " #! Original Equation ! " ! & Rewrite with like bases '% Property of exponential equations Subtract 2 from both sides The solution is 1. Check this in the original equation. b.$% Original Equation $% Property of logarithmic equations ! Add 3 to both sides ( Divide both sides by 2 The solution is 7. Check this in the original equation. Example 2: Solve )' "* . Solution: ) ' "* Original Equation "* ) Subtract 5 from both sides "* ) Take the logarithm of both sides ' ) Inverse Property $')+,(- Subtract 1 from both sides Check: )' "* Original Equation )' *,./0"* Substitute 1.708 for )' ,./0 Simplify )'!,111+ Solution checks |
Solving exponential and logarithmic equations worksheet with answers pdf
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