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Logarithmic function, equation and inequality

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Felina Victoria

This document provides examples and explanations of logarithmic functions, equations, and inequalities. It begins by defining logarithms and showing how to write expressions in logarithmic form. Examples are provided of writing expressions as logarithms and their exponential forms. The document then demonstrates how to solve logarithmic equations by setting the logarithmic expressions equal and solving. It also shows how to evaluate logarithmic expressions and solve equations with logarithmic expressions on both sides.

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LOGARITHMIC FUNCTION, LOGARITHMIC EQUATION AND LOGARITHMIC INEQUALITY FELINA E. VICTORIA MAED-MATH
Example 1: Solution: log2 8 3 We read this as: the log base 2 of 8 is equal to 3. 3 Write 2 8 in logarithmic form.
Example 1a: Write 42 16 in logarithmic form. Solution: log4 16 2 Read as: the log base 4 of 16 is equal to 2.
Example 1b: Solution: Write 2 3 1 8 in logarithmic form. log2 1 8 3 1 Read as: "the log base 2 of is equal to -3". 8
Okay, so now its time for you to try some on your own. 1. Write 72 49 in logarithmic form. 7Solution: log 49 2
log5 1 0Solution: 2. Write 50 1 in logarithmic form.
3. Write 10 2 1 100 in logarithmic form. Solution: log10 1 100 2
Solution: log16 4 1 2 4. Finally, write 16 1 2 4 in logarithmic form.
Example 1: Write log3 81 4 in exp onential form Solution: 34 81
Example 2: Write log2 1 8 3 in exp onential form. Solution: 2 3 1 8
Okay, now you try these next three. 1. Write log10 100 2 in exp onential form. 3. Write log27 3 1 3 in exp onential form. 2. Write log5 1 125 3 in exp onential form.
1. Write log10 100 2 in exp onential form. Solution: 102 100
2. Write log5 1 125 3 in exp onential form. Solution: 3 1 5 125
3. Write log27 3 1 3 in exp onential form. Solution: 27 1 3 3
Solution: Lets rewrite the problem in exponential form. 62 x Were finished ! 6Solve for x: log 2x Example 1
Solution: 5 y 1 25 Rewrite the problem in exponential form. Since 1 25 5 2 э 駈 醐 件 5y 5 2 y 2 5 1 Solve for y: log 25 y Example 2
Example 3 Evaluate log3 27. Try setting this up like this: Solution: log3 27 y Now rewrite in exponential form. 3y 27 3y 33 y 3
These next two problems tend to be some of the trickiest to evaluate. Actually, they are merely identities and the use of our simple rule will show this.
Example 4 Evaluate: log7 72 Solution: Now take it out of the logarithmic form and write it in exponential form. log7 72 y First, we write the problem with a variable. 7y 72 y 2
Example 5 Evaluate: 4log 4 16 Solution: 4 log 4 16 y First, we write the problem with a variable. log4 y log4 16 Now take it out of the exponential form and write it in logarithmic form. Just like 23 8 converts to log2 8 3 y 16
Example 1 Solve: log3 (4x 10) log3 (x 1) Solution: Since the bases are both 3 we simply set the arguments equal. 4x10 x1 3x101 3x 9 x 3
Example 2 Solve: log8 (x2 14) log8 (5x) Solution: Since the bases are both 8 we simply set the arguments equal. x2 14 5x x2 5x 14 0 (x 7)(x 2) 0 Factor (x 7) 0 or (x 2) 0 x 7 or x 2 continued on the next page

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Logarithmic function, equation and inequality

  • 1. LOGARITHMIC FUNCTION, LOGARITHMIC EQUATION AND LOGARITHMIC INEQUALITY FELINA E. VICTORIA MAED-MATH
  • 2.
  • 3. Example 1: Solution: log2 8 3 We read this as: the log base 2 of 8 is equal to 3. 3 Write 2 8 in logarithmic form.
  • 4. Example 1a: Write 42 16 in logarithmic form. Solution: log4 16 2 Read as: the log base 4 of 16 is equal to 2.
  • 5. Example 1b: Solution: Write 2 3 1 8 in logarithmic form. log2 1 8 3 1 Read as: "the log base 2 of is equal to -3". 8
  • 6. Okay, so now its time for you to try some on your own. 1. Write 72 49 in logarithmic form. 7Solution: log 49 2
  • 7. log5 1 0Solution: 2. Write 50 1 in logarithmic form.
  • 8. 3. Write 10 2 1 100 in logarithmic form. Solution: log10 1 100 2
  • 9. Solution: log16 4 1 2 4. Finally, write 16 1 2 4 in logarithmic form.
  • 10. Example 1: Write log3 81 4 in exp onential form Solution: 34 81
  • 11. Example 2: Write log2 1 8 3 in exp onential form. Solution: 2 3 1 8
  • 12. Okay, now you try these next three. 1. Write log10 100 2 in exp onential form. 3. Write log27 3 1 3 in exp onential form. 2. Write log5 1 125 3 in exp onential form.
  • 13. 1. Write log10 100 2 in exp onential form. Solution: 102 100
  • 14. 2. Write log5 1 125 3 in exp onential form. Solution: 3 1 5 125
  • 15. 3. Write log27 3 1 3 in exp onential form. Solution: 27 1 3 3
  • 16. Solution: Lets rewrite the problem in exponential form. 62 x Were finished ! 6Solve for x: log 2x Example 1
  • 17. Solution: 5 y 1 25 Rewrite the problem in exponential form. Since 1 25 5 2 э 駈 醐 件 5y 5 2 y 2 5 1 Solve for y: log 25 y Example 2
  • 18. Example 3 Evaluate log3 27. Try setting this up like this: Solution: log3 27 y Now rewrite in exponential form. 3y 27 3y 33 y 3
  • 19. These next two problems tend to be some of the trickiest to evaluate. Actually, they are merely identities and the use of our simple rule will show this.
  • 20. Example 4 Evaluate: log7 72 Solution: Now take it out of the logarithmic form and write it in exponential form. log7 72 y First, we write the problem with a variable. 7y 72 y 2
  • 21. Example 5 Evaluate: 4log 4 16 Solution: 4 log 4 16 y First, we write the problem with a variable. log4 y log4 16 Now take it out of the exponential form and write it in logarithmic form. Just like 23 8 converts to log2 8 3 y 16
  • 22. Example 1 Solve: log3 (4x 10) log3 (x 1) Solution: Since the bases are both 3 we simply set the arguments equal. 4x10 x1 3x101 3x 9 x 3
  • 23. Example 2 Solve: log8 (x2 14) log8 (5x) Solution: Since the bases are both 8 we simply set the arguments equal. x2 14 5x x2 5x 14 0 (x 7)(x 2) 0 Factor (x 7) 0 or (x 2) 0 x 7 or x 2 continued on the next page