Algebra-2 Logarithmic and Exponential Word Problems

Quiz 7-6: Solve : Use the “power” of the power property for logarithms 1. “Isolate the power” “Undo the power” “Undo the log” 4. Use the “power” of the power property for logarithms

Solving an Exponential Equation: The easiest problem “Isolate the power, Undo the power” What is the inverse function of power base 2? of power base 2? Logarithm exponent property Remember:

Solving by Graphing. Set equal to zero. Replace zero with ‘y’ Solution: the value(s) of ‘x’ that make y =0  x-intercepts!! Graph on calculator Use the “power of the calculator” to find the x-intercept.

Solving Logarithmic equations by graphing x = 10 Set equal to zero (-2 each side) Replace ‘0’ with ‘y’ Solution: the x-value that causes ‘y’ to equal ‘0’, x-intercepts graph Adjust window

Your turn: Solve by graphing : 2. 1.

Your turn: 3. Graph the following two points on an x-y coordinate system: (100, 5) and (50, 3) On the same x-y plot, graph following two points: (1, 5), (2, 3) 4. Why was it difficult to plot the last two points? 5.

Logarithmic Scale (100, 5) (1, 5) (50, 3) (2, 3)

(100, 9.2) (1, 0) (50, 3) (2, 1.4)

What are logs used for in the real world? Anytime you need to measure a quantity that can range over a VERY BIG spread of values. Sound Intensity (watts/square meter) Threshold of hearing: Instant eardrum damage: We convert this to decibels.

Barking dog sound intensity: watts/sq meter Sound Intensity Loudness of the sound (in decibels) as a function (in decibels) as a function of the sound intensity of the sound intensity Intensity of the sound in the sound in watts/sq meter watts/sq meter Intensity of sound at the threshold at the threshold of hearing ( of hearing ( watts per sq meter) watts per sq meter) How Loud is a dog’s bark? (in decibels)

An ambulance’s siren is 120 decibels loud. What is the sound intensity of the siren? Sound Intensity

Your turn: 6. The front row of a rock concert has a sound intensity of of The reference sound intensity The reference sound intensity What is the sound level in decibels on the front row of the rock concert?

7. The sound of rush hour traffic at the Chevron is 70 dB loud. What is the sound intensity? Sound Intensity

Typical sound intensities SourceIntensity Intensity Level Threshold of Hearing (TOH)………….1* W/m 2 0 dB Rustling Leaves………………………..1* W/m 2 ?? dB Whisper…………………………………1* W/m 2 20 dB Normal Conversation………………….1*10 -6 W/m 2 ?? dB Busy Street Traffic……………………..1*10 -5 W/m 2 70 dB Vacuum Cleaner……………………….1*10 -4 W/m 2 ?? dB Large Orchestra………………………..6.3*10 -3 W/m 2 98 dB Walkman at Maximum Level………….1*10 -2 W/m dB Front Rows of Rock Concert………….1*10 -1 W/m 2 ?? dB Threshold of Pain………………………1*10 1 W/m dB Military Jet Takeoff……………………..1*10 2 W/m 2 ??? dB Instant Perforation of Eardrum……….1*10 4 W/m dB

What are logs used for in the real world? Anytime you need to measure a quantity that can range over a VERY BIG spread of values. Measure of the acidity of a solution. Measure of the acidity of a solution. Acidity caused by the amount of “hydronium” ion in solution.  concentration of the hydronium ion. Smallest detectable concentration: Maximum possible concentration: We convert this to pH.

pH Acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H + ]. The measure of acidity used is pH, (the negative of the common log of the hydrogen-ion concentration): pH = -log [H + ] pH = -log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

pH A solution has a pH of 2.3. What is the hydrogen- ion concentration? pH = -log [H+] pH = -log [H+]

pH A solution has hydrogen-ion concentration of moles/liter. What is the pH of the solution? pH = -log [H+] pH = -log [H+] x = -log

Your turn A solution has a pH of 7.9. What is the hydrogen- ion concentration? pH = -log [H+] pH = -log [H+] 7.9 = -log [H+] 8.

Your turn: A solution has hydrogen-ion concentration of moles/liter. What is the pH of the moles/liter. What is the pH of thesolution? pH = -log [H+] pH = -log [H+] 9.

Earthquake Magnitude

MagnitudeEarthquake EffectsEstimated Number Each Year 2.5 or less Usually not felt, but can be recorded by seismograph. 900, to 5.4Often felt, but only causes minor damage.30, to 6.0 Slight damage to buildings and other structures to 6.9 May cause a lot of damage in very populated areas to 7.9Major earthquake. Serious damage or greater Great earthquake. Can totally destroy communities near the epicenter. One every 5 to 10 years Earthquake Magnitude Scale

Earthquake Intensity (Richter Scale) Is measured by the amplitude of the vibration felt at the measuring station. felt at the measuring station. “B” is a “fudge factor” to account for weakening of the seismic wave from the location of the earthquake to the location of where it is measured. Amplitude is measured in

Earthquake Intensity How intense was an earthquake that had the following measurements at the recording station? Amplitude =

Earthquake Intensity You are checking the reported intensity of an earthquake. The station reported an intensity of 7.9. The “fudge factor” for the recording station is 2.5 (B=2.5). What should be the measured amplitude of the strip chart recorder?

Your turn: 10. What is the richer scale intensity of an earthquake that was measured at a recording station that has a “fudge factor” of 2.9 (B = 2.9)? The actual meter deflection had an amplitude of 750 ɥ m.

Real World Logarithmic Model Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. In 1925, a tornado traveled 220 miles through 3 states. Estimate the wind speed inside the tornado. (1) Plug numbers into the formula (2) Solve for the unknown variable in the formula

Real World Logarithmic Model What if the problem was: “The wind speed is 200mph, how far will the tornado travel on the ground? “The wind speed is 200mph, how far will the tornado travel on the ground? (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula (remember: “isolate the log, undo the log”)

Real World Logarithmic Model “The wind speed is 200mph, how far will the tornado travel on the ground? Subtract 65 from both sides Divide by 93 (both sides) remember: “isolate the radical” ?  same thing, “isolate the log” We need the “inverse function” of log (base 10). log (base 10). Exponential base 10 We will solve log and exponential equations later.

Your turn: Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. 11. A tornado traveled 100 miles. Estimate the wind speed inside the tornado. (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula

Your turn: Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. 12. Some storm chasers measured the speed of the wind inside a tornado. It was 275 mph. How far will the tornado travel along the ground?. (1) Plug numbers into the formula (2) Solve for the unknown variable in the formula

Population Growth Population (as a function of time) function of time) Initial population population Growth rate rate time It’s just a formula!!! The initial population of a colony of bacteria is The population increases by 50% is The population increases by 50% every hour. What is the population after 5 hours? every hour. What is the population after 5 hours? Percent rate of change (in decimal form) (in decimal form)

Simple Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”)Interest rate rate time A bank account pays 3.5% interest per year. If you initially invest $200, how much money If you initially invest $200, how much money will you have after 5 years? will you have after 5 years?

Your turn: A bank account pays 14% interest per year. If you initially invest $2500, how much money If you initially invest $2500, how much money will you have after 7 years? will you have after 7 years? The population of a small town was 1500 in 1989 The population increases by 3% every The population increases by 3% every year. What is the population in 2009? year. What is the population in 2009?

Your turn: A bank account earning 4% interest has $3569 in it. The original deposit was $2000. How long has the money been in the account? 15.

Newton’s Law of Cooling A high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be modeled by the following equation. Temperature (as a function of time) Temperature of the medium the medium Initial Temp of the object of the object Constant, determined by the heat transfer by the heat transfer characteristics of the material characteristics of the material Time

Newton’s Law of Cooling The temperatur of a cake just out of the oven is 350°. After 15 minutes it cools to 125°. The temperature of the room is 70°. How long did it take to cool to 200° 15 minutes it cools to 125°. The temperature of the room is 70°. How long did it take to cool to 200° (1) Find the cooling factor ‘k’ for the given conditions. Using Newton’s law of cooling to find cooling time is a 2 step problem. problem. “Isolate the Power, undo the power”

Newton’s Law of Cooling The temperatur of a cake just out of the oven is 350°. After 15 minutes it cools to 125°. The temperature of the room is 70°. How long did it take to cool to 200° 15 minutes it cools to 125°. The temperature of the room is 70°. How long did it take to cool to 200° (1) Find the cooling factor ‘k’ for the given conditions. “Isolate the Power, undo the power” (2) Use ‘k’ to find the time to cool to 200° F.

Your turn: 16. A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

Approximating Expressions If you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms: values and properties of logarithms to find other logarithms: Use these to find: Use your calculator to find: log 2 = ?

Approximating Expressions If you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms: values and properties of logarithms to find other logarithms: Use these to find: Use your calculator to find: log 1/2 = ?

Approximating Expressions If you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms: values and properties of logarithms to find other logarithms: Use these to find: Use your calculator to find: log 18 = ?

Your turn: Don’t use your calculator: Use the following to answer the questions. to answer the questions