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# 7.6 solving logarithmic equations

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• 1. Exponential & Logarithmic Equations • Exponential Equations with Like Bases • Exponential Equations with Different Bases • Logarithmic Equations
• 2. Exponential Equations with Like Bases In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like… If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple. 32x 1 5 4 or 3x 1 9x 2
• 3. Exponential Equations with Like Bases Example #1 - One exponential expression. 32x 1 5 4 32x 1 9 32x 1 32 1. Isolate the exponential expression and rewrite the constant in terms of the same base. 2x 1 2 2x 1 x 1 2 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.
• 4. Exponential Equations with Like Bases Example #2 - Two exponential expressions. 3x 1 9x 2 3x 1 32 x 2 3x 1 32x 4 1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation. x 1 2x 4 x 5
• 5. Change-of-Base Formula The base you change to can be any base so generally we’ll want to change to a base so we can use our calculator. That would be either base 10 or base e. LOG “common” log base 10 LN “natural” log base e a M log log a M ln ln Example for TI-83 If we generalize the process we just did we come up with the: a M M b b a log log log
• 6. Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places. 16log3 Since 32 = 9 and 33 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3. 3ln 16ln 16log3 put in calculator 524.2
• 7. Exponential Equations with Different Bases The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number. The solutions are irrational numbers, we will need to use a log function to evaluate them. 32x 1 5 11 or 3x 1 4x 2
• 8. Exponential Equations with Different Bases Example #1 - One exponential expression. 1. Isolate the exponential expression. 3. Use the log rule that lets you rewrite the exponent as a multiplier. 32x 1 5 11 32x 1 16 2. Take the log (log or ln) of both sides of the equation.ln 32x 1 ln 16 (2x 1)ln3 ln16
• 9. Exponential Equations with Different Bases Example #1 - One exponential expression. 4. Isolate the variable. 2x 1 ln16 ln 3 2x ln16 ln 3 1 x ln16 2ln 3 1 2 x 0.762 (2x 1)ln3 ln16
• 10. Exponential Equations with Different Bases Example #2 - Two exponential expressions. 1. The exponential expressions are already isolated. 3. Use the log rule that lets you rewrite the exponent as a multiplier on each side.. 2. Take the log (log or ln) of both sides of the equation. 3x 1 4x 2 ln 3x 1 ln 4x 2 (x 1)ln3 (x 2)ln4
• 11. Exponential Equations with Different Bases Example #2 - Two exponential expressions. 4. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide. xln3 ln3 xln4 2ln4 xln3 xln4 ln3 2ln4 x(ln3 ln4) (ln3 2ln4) x (ln3 2ln4) ln3 ln4 x 13.457
• 12. Logarithmic Equations In a Logarithmic Equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example … log4 2x 1 3 5 lnx ln(2x 1) 1 logx 3 2
• 13. Logarithmic Equations Example 1 - Variable inside the log function. log4 2x 1 3 5 log4 2x 1 2 42 2x 1 16 2x 1 2x 17 x 8.5 1. Isolate the log expression. 2. Rewrite the log equation as an exponential equation and solve for ‘x’.
• 14. Logarithmic Equations Example 2 - Variable inside the log function, two log expressions. ln x ln(2x 1) 1 ln x 2x 1 1 e1 x 2x 1 e(2x 1) x 2ex e x 2ex x e x(2e 1) e x e 2e 1 1. To isolate the log expression, we 1st must use the log property to combine a difference of logs. 2. Rewrite the log equation as an exponential equation (here, the base is ‘e’). 3. To solve for ‘x’ we must distribute the ‘e’ and then collect the ‘x’ terms together and factor out the ‘x’ and divide. x 0.613
• 15. Logarithmic Equations Example 3 - Variable inside the base of the log. logx 3 2 x 2 3 x 2 1 2 3 1 2 x 1 3 x 3 3 1. Rewrite the log equation as an exponential equation. 2. Solve the exponential equation.