PROPERTIES OF LOGARITHMIC FUNCTIONS EXPONENTIAL FUNCTIONS An exponential function is a function of the form f (x) = b x , where b > 0 and x is any real number. (Note that f (x) = x 2 is NOT an exponential function.) LOGARITHMIC FUNCTIONS = = y > > ≠ log b x y means that x b where x ,0 b ,0 b 1 Think: Raise b to the power of y to obtain x. y is the exponent. The key thing to remember about logarithms is that the logarithm is an exponent! The rules of exponents apply to these and make simplifying logarithms easier. = = 2 Example: log 10 100 2 , since 100 10 . log 10 x is often written as just logx , and is called the COMMON logarithm. ≈ loge x is often written as ln x , and is called the NATURAL logarithm (note: e .2 7182818284 59 ... ). PROPERTIES OF LOGARITHMS EXAMPLES = + + = = 1. log b MN log b M log b N log 50 log 2 log 100 2 Think: Multiply two numbers with the same base, add the exponents. M 56 2. log = log M − log N log 56 − log 7 = log = log 8 = 1 b N b b 8 8 8 7 8 Think: Divide two numbers with the same base, subtract the exponents. P = 3 = ⋅ = ⋅ = 3. log b M P log b M log 100 3 log 100 3 2 6 Think: Raise an exponential expression to a power and multiply the exponents together. x = = 0 = = log b b x log b 1 0 (in exponential form, b 1) ln 1 0 = = = log b b 1 log 10 10 1 ln e 1 x = x = x = log b b x log 10 10 x lne x log b x = = b x Notice that we could substitute y log b x into the expression on the left y = to form b . Simply re-write the equation y log b x in exponential form as x = b y . Therefore, b log b x = b y = x . Ex: eln 26 = 26 CHANGE OF BASE FORMULA log N = a = log 5 ≈ .0 698970 ≈ log b N , for any positive base a. log 12 5 .0 6476854 log a b log 12 .1 079181 This means you can use a regular scientific calculator to evaluate logs for any base. Practice Problems contributed by Sarah Leyden, typed solutions by Scott Fallstrom Solve for x (do not use a calculator). ( 2 − ) = x = 2 − ( + ) = 1. log 9 x 10 1 6. log 3 27 5.4 10. log 2 x log 2 3x 8 1 2. log 32x+1 = 15 3 3 7. log 8 = − x 11. (1 )log x − (1 )log x 2 = 1 3. log 8 = 3 2 2 3 3 3 x + ( − ) = = 8. log 6 x log 6 x 1 1 4. log 5 x 2 1 1 2 2 + ( ) = ( − + ) = 9. log 2 x log 2 x 3 5. log 5 x 7x 7 0 Solve for x, use your calculator (if needed) for an approximation of x in decimal form. 12. 7 x = 54 15. 10x = e 18. 8 x = 9 x = − x = x+1 = 4 13. log 10 x 17 16. e 7.1 19. 10 e x x ( ) = = − 14. 5 = 9 ⋅ 4 17. ln ln x .1 013 20. log x 10 54.1 Solutions to the Practice Problems on Logarithms: ( 2 − ) = ⇒ 1 = 2 − ⇒ 2 = ⇒ = ± 1. log 9 x 10 1 9 x 10 x 19 x 19 2x+1 = ⇒ 15 = 2x+1 ⇒ + = ⇒ = ⇒ = 2. log 3 3 15 3 3 2x 1 15 2x 14 x 7 = ⇒ 3 = ⇒ = = ⇒ 2 = ⇒ = 3. log x 8 3 x 8 x 2 4. log 5 x 2 5 x x 25 ( 2 − + ) = ⇒ 0 = 2 − + ⇒ = 2 − + ⇒ = ( − )( − ) ⇒ = = 5. log 5 x 7x 7 0 5 x 7x 7 0 x 7x 6 0 x 6 x 1 x 6 or x 1 x = ( 3 )x = 3x = = = 6. log 3 27 5.4 ⇒ log 3 3 5.4 ⇒ log 3 3 5.4 ⇒ 3x 5.4 ⇒ x 5.1 − 3 − 2 = − 3 ⇒ 2 = ⇒ = 3 ⇒ = 1 7. log x 8 2 x 8 x 8 x 4 + ( − ) = ⇒ ( 2 − ) = ⇒ 2 − = ⇒ 2 − − = ⇒ log 6 x log 6 x 1 1 log 6 x x 1 x x 6 x x 6 0 8. (x − 3 )(x + 2 ) = 0 ⇒ x = 3 or x = − .2 Note : x = −2 is an extraneous solution, which solves only the new equation. x = 3 is the only solution t o the original equation. 1 2 1 1 x − 1 − 1 −2 2 + = ⇒ = ⇒ 2 = ⇒ 3 = 2 ⇒ = ()3 = 1 9. log 2 x log 2 3 log 2 3 log 2 x 3 2 x x 2 64 x x log x 2 − log (3x + 8) = 1 ⇒ log ( x2 ) = 1 ⇒ x2 = 2 ⇒ x 2 = 6x +16 ⇒ 10. 2 2 2 3x+8 3x+8 x 2 − 6x −16 = 0 ⇒ (x − 8 )(x + 2 ) = 0 ⇒ x = 8 or x = −2 1 2 1 2 x 1 −2 ()1 − ()1 2 = ⇒ 2 − 3 = ⇒ = ⇒ 2 3 = ⇒ 2 log 3 x 3 log 3 x 1 log 3 x log 3 x 1 log 3 2 1 x 3 11. x 3 − 1 − 6 = ⇒ = 6 = 1 x 3 x 3 729 log 54 12. 7 x = 54 ⇒ x = log 54 ⇒ x = ≈ .2 0499 7 = ⇒ = 17 log 7 13. log 10 x 17 x 10 x = ⋅ x ⇒ 5x = ⇒ (5 )x = ⇒ = ⇒ ≈ 14. 5 9 4 x 9 4 9 x log 5 9 x .9 8467 4 4 x = ⇒ = ⇒ = ≈ 15. 10 e x log 10 e x log e .0 4343 16. e− x = 7.1 ⇒ −x = ln 7.1 ⇒ x = −ln 7.1 ≈ − .0 5306 .1 013 17. ln (ln x) = .1 013 ⇒ ln x = e .1 013 ⇒ x = e e ≈ 15 .7030 x = x ⇒ = (9 )x ⇒ = ⇒ = 18. 8 9 1 8 x log 9 1 x 0 8 x+1 = 4 ⇒ + = 4 ⇒ = 4 − = 4 − ⇒ = (e4 ) ≈ 19. 10 e x 1 log e x log e 1 log e log 10 x log 10 .0 7372 − − 1 = − ⇒ 54.1 = ⇒ = 54.1 ≈ 20. log x 10 54.1 x 10 x 10 .0 2242 .