Solving Equations using

LogarithmsPart 1: The process/ procedure

Part 2: The application

RecallThe human population (over a limited data range, of course), can be modeled by an exponential growth equation of the form:

In an earlier lesson, we did a trial and error approach to find the time it will take to hit a given population,

Using logs, we now have a strategy for getting an exact answer.

Example1: Solve𝟓𝟑 𝒙−𝟏=𝟐𝟗

Just like with any equation we want to get x by itself:

Once we isolate the exponent, we can multiply bothSides by ‘log’ to solve for the exponent:

Apply the product rule of logs:

Divide both sides to get x by itself:

Simplify:

Check:

+1 +1𝟓𝟑 𝒙=𝟑𝟎

𝒍𝒐𝒈𝟓𝟑 𝒙=𝒍𝒐𝒈𝟑𝟎3xlog5=log303log5 3log5

x≈0.7044𝟓𝟑∗ .𝟕𝟎𝟒𝟒−𝟏=𝟐𝟖 .𝟗𝟒 (𝒄𝒍𝒐𝒔𝒆𝒆𝒏𝒐𝒖𝒈𝒉)

Example2: SolveSince the quantity in parenthesis is multiplied by 5, we can divide both sides by 5:

Multiply both sided by log and apply the product property of log:

Divide both sides by log6 and simplify:

(You do) Solve for x and plug into the original equation to check

5 5

𝟔𝟐 𝒙−𝟑=𝟐𝟎

→𝒍𝒐𝒈𝟔 𝒍𝒐𝒈𝟔

→𝟐𝒙 −𝟑=𝟏 .𝟔𝟕𝟐

𝟔𝒙−𝟏

𝟓 −𝟑=𝟗

Example3: SolveLook at the graph: What is an accurate guess you can make for what x SHOULD equal? Why?

Ladies do: Gentlemen do:

x≈0.675 x≈4.085

Round all answers to THREE Decimal Places

Two option: 1) Find someone who

got the answer and work with them to make sure you can get it as well

2) Find someone who DID not get the answer and work with them to make sure they can get it.

Application with Logs

Measuring sound

0 dB → Threshold of hearing

20 dB → Whispering

60 dB → Normal Conversation

80 dB → Vacuum Cleaner

110dB → Front row at a rock concert

130 dB → Threshold of pain

160 dB → Bursting eardrums

Sound level chart Two properties of a sound wave:

1) Intensity measures how much energy is in the wave

2) Decibel level measures how loud we hear the sound and is related to the intensity of the wave.

EquationsTwo sound-related equations we will use:

• I = Intensity of sound measured in • P = Power of sound source measured in watts• r = radius from the sound source measured in meters

2)

You bought a 500 watt speaker for your car and play your favorite Justin Bieber CD on it while sitting at a distance of 1 meter away from the speaker. Will it cause permanent hearing damage (ignoring the fact it’s a Justin Bieber CD)?

1) Find the intensity of the sound wave:

2) Substitute the intensity into the equation:

≈146 dB

So, long exposure to it will cause hearing loss but it won’t burst your eardrums (which happens at 160dB)

If a 500 watt speaker is not enough, how powerful of a speaker will you need to have in order to burst your eardrums?

Goal: Solve for P when dB =160 (loud enough for eardrums to burst)

Note: Since in the equation you don’t know ‘P’ or ‘I’ you can’t use it YET.

However, you can use the dB level equation to find I:

160

Divide → 16 =

Use the properties of logs → 16 =

= 12 → 16 = 12 + log

Subtract → 4 = log

Rewrite in exponential form: = 10000

Solve for P:

10000

P ≈ 125,663 Watt speaker you would need to produce a sound at 160dB.

How loud does a 150 watt speaker sound from 100 meters away?

Know:• Power = 150 W• r = 100 m

Need to know: sound Intensity (I) and sound level (dB)

• Which equation should be used first: because we know P and r and want to find I

≈ 111 dBequivalent of front row at rock concert

The current world population is 7.41 Billion people. Suppose the table below is accurate and the population grows at 1.13% per year. If now it is 2016, what year will the population:

1) Reach 8 Billion

2) Reach 10 billion

Exponential growth function:

Your TaskWrite and illustrate an application problem with logs. You need to think of a creative problem that is more then just taking my example and changing number around. You final product will be on an 8.5x11 computer paper and organized like this:

Written problem

Solution

Illustration