# Unit 5.4 Exponential And Logarithmic Equations From Previous Units

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Unit 5.4 – Exponential and Logarithmic EquationsFrom previous units

Exponent Rules1) Zero-ExponentAny number (excluding 0) to the 0 poweris always equal to 1. Example: x0 12) Zero-To-Exponent0 to any power is 0, except 00, which is undefined.3) One-ExponentAny number to the 1st power is alwaysequal to that number. Example: x1 x4) Product Of PowersWhen a number to a power is multiplied by thesame number to a power, add the powers.Example: 𝑥 𝑚 𝑥 𝑛 𝑥 𝑚 𝑛5) Quotient Of PowersWhen a number to a power is divided by thesame number to a power, subtract the powers.Example:𝑥𝑚𝑥𝑛 𝑥 𝑚 𝑛6) Power of a PowerWhen a number to a power is raised to anotherpower, multiply the powers.Example: (𝑥 𝑚 )𝑛 𝑥 𝑚 𝑛7) Negative ExponentAny number raised to a negative power is equalto 1 divided by the number raised to a positive1power. Example: 𝑥 7 7𝑥8) Power of a ProductWhen two numbers are multiplied and raisedto a power, it’s equal to both numbers individuallyraised to that power and multiplied together.Example: (𝑥𝑦)𝑚 𝑥 𝑚 𝑦 𝑚9) Power of a QuotientWhen two numbers are divided and raised to apower, it’s equal to both numbers individuallyraised to that power and divided.Example:𝑥( )4𝑦 𝑥4𝑦4

Reminder:Similar to3642 2643

Specificallyax ay if x yln x ln y if x yax ay if x y𝑙𝑛 𝑒 𝑥 𝑥CW # 1a) 2𝑥 512Solve each equation for x. Which strategy did you use?b) log 6 𝑥 3c) 5 𝑒 𝑥 0d) 9𝑥 13

The next 4 slides involve solving EXPONENTIAL equations. Approximate the result to three decimal places,if necessary.Strategy #1It is recommendedto check your solutionsin the original equation.Strategy #2

Strategy #2𝑙𝑛 𝑒 𝑥 𝑥orIt is recommendedto check your solutionsin the original equation.x ln e ln 55x ln 55CW # 2Solve the equations and approximate the results to three decimal places,if necessary. Which strategies are used for each?c.

Strategy #2It is recommendedto check your solutionsin the original equation.CW # 3Solve and approximate the results to three decimal places.Which strategy is used?

𝑒𝑥 2Convert to logarithmic form𝑙𝑛𝑒𝑥 1CW # 4𝑒𝑥 𝑥It is recommendedto check your solutionsin the original equation.

The next 4 slides involve solving LOGARITHMIC equations. We start with Strategy #1.It is recommendedto check your solutionsin the original equationto verify that the answeris correct, and to makesure that the answer is inthe domain of theoriginal equation.

CW # 5Solve each equation.Strategy #3It is recommended to check yoursolutions in the original equation toverify that the answer is correct, andto make sure that the answer is inthe domain of the original equation.𝑥 11𝑒21 𝑒

Strategy #3It is recommended to check yoursolutions in the original equation toverify that the answer is correct, andto make sure that the answer is inthe domain of the original equation.CW # 6a)b)

The solutions appear to be x –4and x 5. However, when youcheck these in the originalequation, you can see that x 5 isthe only solution.One way to tell: In order to takethe logarithm of a number, thatnumber must be positive. So aboveleft, for log 5x and log (x – 1),both 5x and (x – 1) must begreater than zero. Meaning, ananswer of –4 can’t work.CW # 7

CW # 8

CW # 9In the previous example, during which year did the sales reach 180 billion?

6) Power of a Power When a number to a power is raised to another power, multiply the powers. Example: ( à) á à á 7) Negative Exponent Any number raised to a negative power is equal to 1 divided by the number raised to a positive power. Example: 7 1 ë7 8) Power of a Product When two numbers are multiplied and raised

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