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# Real World Problems Using Logarithmic & Exponential Equations......

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#### Transcript of Real World Problems Using Logarithmic & Exponential Equations......

Real Life World Problems Using Logarithmic & Exponential Equations.

By: Ikram , Faria &Fatmeh Example 1 : Step 1: Substitute the given data into the equation:

5.5 = 100(1/2)^12/h

Step 2: Isolate 'h' by dividing both sides by the value in front of (1/2)^16/h (100). 5.5 /100

Step3: Express the left side of the equationas a fraction.

0.05 = (1/2)^12/h

Step 4: Express both sides with fractions as decimals.

0.055 = (0.5)^12/h

Step 5: If the 2 quantities are equal than the logs of the quantities will also be equal. Apply log to both sides.

log (0.055) = log (0.5)^12/h

Step 6: Use the power rule for lograithms to rewrite the right side of the equation without an exponents.

log(0.055) = 12/h log(0.5)

Step7: Mutlply both sides by 'h'

h log (0.055) = 12 log(0.5)

Step8:Divide both sides by log(0.055)

h = 12log(0.5)/log(0.055)

Step9: Evaluate with a calculator

h = 2.9 days

there for h = 3 days Geologists has estimated that the earthquake that has occurred in Australia has been measured to be 9.1 on the Richter scale, and the period of the seismic wave was 2.1s. If the weakening of the earthquake waves equal to 8.4, what was rthe amplitude a , of the vertical grand motion. Radioactive scientists work in laboratory, they have received a shipment of 100g of radioactive, and 12days later 5.5g of uranium are left remained after using it in an experiment . What is the half-life of uranium? Intorduction:

An exponential function is a mathematical expression in which a variable represents the exponent of an expression. Example of Exponential :

Solve for x: 4=2

This one is actually pretty simple, so let's just think it through:

The problem says we have to multiply x number of two's together to get four. Well, everyone knows that 2*2=4, so the answer is two:

4=2

X=2 1. 2. Logarithmic function is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents.

1. Expand

log 7xas the sum of 2 logarithms. Using the first law given above, our answer is

log 7x = log 7 + log x

This has the same meaning as 10 x 10 = 10

this question is not the same as log , which "loh of x to the base "7", which is quite different. x 7 + x Example of Logarithmic: Soluation The general formala to use is A = A (1/2)^ f 0 t / h A- respreasents remaining mass.

f- respreasents the finall amount.

A - respreasnts the intial amount (orginal amount).

t - respreasent time take of a sample.

h- respreasents the half-life period. 0 7 7 + x Example 2: The general formal R = log (a/T)+B

a- the amplitude of the verticle ground motions in microns (U).

T= the period of the seismic waves in secand

B is a factor that accounts for the weaknening of seismic waves.

(1 U is equivatent to 10 m)

Determine the amplitude a , of the vertical ground motion? -6 Soultion : step 1: determine what is given in the problem

R = 9.1

T = 2.1s

B = 8.4

a = ?

Step 2: Write the formula and substitute the given values into the equation.

R = log (a/T)+B

9.1= log (a/2.1)+8.4

Step3: Isolate the term with unknown 'a' by subtracting 8.4 from both sides

9.1-8.4 = log(a/2.1)

0.7 = log(a/2.1)

Steps5: Multiply both sides by 2.1 to solve for "a".

10 x 2.1 = a

10.52 u = a

Step6: State the conclution.

The amplitude of the vertical ground motion was about 10.52 u ( about 0.010520 cm). 0.7 logarithmic Example 3: Kevin drinks a coffee at Tim Hortons with the initial amount of 100mg to 150mg, the half life of coffeine is 1.55h. 3 hours passed since kevin drank coffee, How many coffeine did he intake?

The appropriate formula used is y=c(0.5)t/h

y = the final amount of coffeine

c = te initial amount of coffeine

t = the time taken of sample

h = is the half-life of the coffeine Solution: y = c(0.5)t/h

y = 100(0.5)3/1.55

y =100(0.5)^2

y = 25mg

Therefore, 25mg of coffeine remain in kevin's bloodstream after 3h. A sound 16 dB is 5 times louder then a weaker sound. what is the loudness of the weaker sound.

The general formula: dB-dB=10 log(I/I).

dB = descerible level tthe loud sound

dB = describle level of soft sound

I/I = Intensity ratio of sound Soluation:

dB-dB=10 log (I/I )

16-dB=10 log(5)

16-dB=6.9897

-16+16-dB=16-6.9897

dB=9.011

Therefore, the loudness of the weaker sound is 9.011 dB . The Rogers company has an investment of $6500 grows at a rate of 8.8% per year, compunded annually. how long will it take for the investment to be worth $8000. The general formula is A = P (1+i)^n where:

A = the final value of the investment

P = the initial amount of the investment

i = the initial rate of the investment (annually)

n = the number of the compouding periods Soluation: A = P(1+i)^n

8000=6500(1+0.088)^n

8000/6500=(1+0.088)^n

1.23=(1+0.088)^n

1.23=(1.088)^n

(Take the log of both sides to solve for n.)

log (1.23)=log(1.088)^n

Divid both sides by log(1.088)^n

n=log(1.23)/log(1.088)

n=2.16 (approximately)

Therefore, it will take approximately 2.16 years for the investment to be worth $8000. Exponential formula Expenontial Form 0 0 0 0 0 0 Example 4: Example 5: EXAMPLE 6: Any Questions??? Earthquake in Australia Earthquake Events of earthquales in Australia What is the Logirathmic function? What is an exponential function? The End Thanks everyone for listening to our awesome

persentation. Sound Waves Expenontial MHF4U Math Project

Full transcriptBy: Ikram , Faria &Fatmeh Example 1 : Step 1: Substitute the given data into the equation:

5.5 = 100(1/2)^12/h

Step 2: Isolate 'h' by dividing both sides by the value in front of (1/2)^16/h (100). 5.5 /100

Step3: Express the left side of the equationas a fraction.

0.05 = (1/2)^12/h

Step 4: Express both sides with fractions as decimals.

0.055 = (0.5)^12/h

Step 5: If the 2 quantities are equal than the logs of the quantities will also be equal. Apply log to both sides.

log (0.055) = log (0.5)^12/h

Step 6: Use the power rule for lograithms to rewrite the right side of the equation without an exponents.

log(0.055) = 12/h log(0.5)

Step7: Mutlply both sides by 'h'

h log (0.055) = 12 log(0.5)

Step8:Divide both sides by log(0.055)

h = 12log(0.5)/log(0.055)

Step9: Evaluate with a calculator

h = 2.9 days

there for h = 3 days Geologists has estimated that the earthquake that has occurred in Australia has been measured to be 9.1 on the Richter scale, and the period of the seismic wave was 2.1s. If the weakening of the earthquake waves equal to 8.4, what was rthe amplitude a , of the vertical grand motion. Radioactive scientists work in laboratory, they have received a shipment of 100g of radioactive, and 12days later 5.5g of uranium are left remained after using it in an experiment . What is the half-life of uranium? Intorduction:

An exponential function is a mathematical expression in which a variable represents the exponent of an expression. Example of Exponential :

Solve for x: 4=2

This one is actually pretty simple, so let's just think it through:

The problem says we have to multiply x number of two's together to get four. Well, everyone knows that 2*2=4, so the answer is two:

4=2

X=2 1. 2. Logarithmic function is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents.

1. Expand

log 7xas the sum of 2 logarithms. Using the first law given above, our answer is

log 7x = log 7 + log x

This has the same meaning as 10 x 10 = 10

this question is not the same as log , which "loh of x to the base "7", which is quite different. x 7 + x Example of Logarithmic: Soluation The general formala to use is A = A (1/2)^ f 0 t / h A- respreasents remaining mass.

f- respreasents the finall amount.

A - respreasnts the intial amount (orginal amount).

t - respreasent time take of a sample.

h- respreasents the half-life period. 0 7 7 + x Example 2: The general formal R = log (a/T)+B

a- the amplitude of the verticle ground motions in microns (U).

T= the period of the seismic waves in secand

B is a factor that accounts for the weaknening of seismic waves.

(1 U is equivatent to 10 m)

Determine the amplitude a , of the vertical ground motion? -6 Soultion : step 1: determine what is given in the problem

R = 9.1

T = 2.1s

B = 8.4

a = ?

Step 2: Write the formula and substitute the given values into the equation.

R = log (a/T)+B

9.1= log (a/2.1)+8.4

Step3: Isolate the term with unknown 'a' by subtracting 8.4 from both sides

9.1-8.4 = log(a/2.1)

0.7 = log(a/2.1)

Steps5: Multiply both sides by 2.1 to solve for "a".

10 x 2.1 = a

10.52 u = a

Step6: State the conclution.

The amplitude of the vertical ground motion was about 10.52 u ( about 0.010520 cm). 0.7 logarithmic Example 3: Kevin drinks a coffee at Tim Hortons with the initial amount of 100mg to 150mg, the half life of coffeine is 1.55h. 3 hours passed since kevin drank coffee, How many coffeine did he intake?

The appropriate formula used is y=c(0.5)t/h

y = the final amount of coffeine

c = te initial amount of coffeine

t = the time taken of sample

h = is the half-life of the coffeine Solution: y = c(0.5)t/h

y = 100(0.5)3/1.55

y =100(0.5)^2

y = 25mg

Therefore, 25mg of coffeine remain in kevin's bloodstream after 3h. A sound 16 dB is 5 times louder then a weaker sound. what is the loudness of the weaker sound.

The general formula: dB-dB=10 log(I/I).

dB = descerible level tthe loud sound

dB = describle level of soft sound

I/I = Intensity ratio of sound Soluation:

dB-dB=10 log (I/I )

16-dB=10 log(5)

16-dB=6.9897

-16+16-dB=16-6.9897

dB=9.011

Therefore, the loudness of the weaker sound is 9.011 dB . The Rogers company has an investment of $6500 grows at a rate of 8.8% per year, compunded annually. how long will it take for the investment to be worth $8000. The general formula is A = P (1+i)^n where:

A = the final value of the investment

P = the initial amount of the investment

i = the initial rate of the investment (annually)

n = the number of the compouding periods Soluation: A = P(1+i)^n

8000=6500(1+0.088)^n

8000/6500=(1+0.088)^n

1.23=(1+0.088)^n

1.23=(1.088)^n

(Take the log of both sides to solve for n.)

log (1.23)=log(1.088)^n

Divid both sides by log(1.088)^n

n=log(1.23)/log(1.088)

n=2.16 (approximately)

Therefore, it will take approximately 2.16 years for the investment to be worth $8000. Exponential formula Expenontial Form 0 0 0 0 0 0 Example 4: Example 5: EXAMPLE 6: Any Questions??? Earthquake in Australia Earthquake Events of earthquales in Australia What is the Logirathmic function? What is an exponential function? The End Thanks everyone for listening to our awesome

persentation. Sound Waves Expenontial MHF4U Math Project