Logarithm lesson
- 1. Simplify- 1) 24 = 2) 1.53 = 3) 31.5 = Solve for x (round to TWO decimal places if you have to) 4) 2 𝑥 = 8 5) 2 𝑥 = 20 6) 3 𝑥 = 100 How did you go about trying to find the answer to #6 and #7?
- 2. Goals: 1) Explain the structure and the purpose of logarithms 2) Solve equations using logarithms
- 3. …In the 11 or 12 years you were at school you were taught the math that took over 6000 years to develop. The development of ‘x’ 1) x+1 = 2 → adding/subtracting 2) 2x=4 → Multiplying/dividing 3) 𝑥2 = 4 → Powers/roots 4) 𝑥 = −1 → Imaginary/complex numbers 5) 2 𝑥 = 5 → exponents/logarithms
- 4. …Created logarithms to make calculating big numbers easier (before electronic calculators) If you need to work with a big messy number like: 123456789.97654321 You could instead say: For what x will 10 𝑥equal the number I want. Since logs follow similar rules as other operations it makes calculating MUCH simpler.
- 5. Magnitude 9.1 earthquake of the coast of Indonesia in 2004, created a tsunami so powerful in sped up the spin of the earth by a fraction of second.
- 6. The explosion of Krakatoa was about 180dB. If you were within 40 miles of the explosion, it would be the last sound you would never hear because the energy from the sound wave would burst your eardrums before you actually heard the sound.
- 7. Just like addition is the inverse of subtraction and multiplication is the inverse of division, Notes start here: Logarithms (or logs) are the inverse of exponents. If 𝑓 𝑥 = 2 𝑥, then logarithms answers the question for what x will the following be true: 𝑥 = 2 𝑓 𝑥
- 8. 𝑓 𝑥 = 2 𝑥 𝑓−1 𝑥 = 𝑙𝑜𝑔2 𝑥
- 9. If 𝑏 𝑥 = 𝑎 then 𝑙𝑜𝑔 𝑏 𝑎 = 𝑥 as long as b > 1, b ≠ 0 Exponent base Log base
- 10. Write the following exponent equation in log form: 1) 52 = 25 2) 6 𝑥 = 100 3) Write your own
- 11. Write the following log equations in exponential form: 1) 𝑙𝑜𝑔232 = 8 2) 𝑙𝑜𝑔4 𝑥 = 20 3) 𝑙𝑜𝑔650 = 𝑥 4) Write your own
- 12. Rewrite in log form, use the change of base formula, solve to THREE decimal places, check: 1) Rewrite in log form: 𝑙𝑜𝑔210 = 𝑥 2) Since most calculators are only able to do log base 10 and log base e, you need to use a change of base formula: 𝑙𝑜𝑔 𝑥 𝑦 = 𝑙𝑜𝑔 𝑦 𝑙𝑜𝑔 𝑥 → 𝑙𝑜𝑔210 = 𝑙𝑜𝑔10 𝑙𝑜𝑔2 3) Solve: x= 𝑙𝑜𝑔210 = 𝑙𝑜𝑔10 𝑙𝑜𝑔2 = 3.322 4) Check: 23.322 = 10 (close enough)
- 13. Solve and check: 5 𝑥 = 15,000 1) Log form → 𝑙𝑜𝑔____________ = x 2) Change of base and solve → 𝑙𝑜𝑔515000 = 𝑙𝑜𝑔____ 𝑙𝑜𝑔____ = _______ 3) Check you answer: 55.975 = 15008 (close enough but if I needed to be more accurate I can always take more decimal places.)