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NATURAL LOGARITHMS
Unit Overview
In this unit you will evaluate natural exponential and natural logarithmic functions and model exponential growth and decay processes. You will also solve logarithmic and exponential equations by using algebra and graphs. Real world problems involving exponential and logarithmic relationships will be solved at the conclusion of the unit.
Natural Exponential and Natural Logarithmic Functions
The number e is an important irrational number. It is approximately equal to 2.71828. Like pi , the value is constant.
An exponential function with the base e is called a natural base exponential function. These functions are useful in describing continuous growth or decay. For example, the number e is used to solve problems involving continuous compound interest and continuous radioactive decay. Exponential functions with the base e have the same properties as other exponential function.
The natural logarithmic function y log (^) ex is abbreviated y ln x and is the inverse of
the natural exponential function y ex.
e 2.
y e^ x y^ x
y ln x
Example #1 : If e^5^ 148.413, what is x in the expression ln x = 5? (^5) 148. ln(148.413) 5 Write the exponential as the inverse logarithm (ln).
e
x
*Note: The three triangular dots are an abbreviation for the word, “therefore”.
Example #2 : If ln 54.598 4 , what is x in the expression e x 54.598?
4
ln 54.598 4 54.598 Write the exponential as the inverse logarithm (ln). 4
e x
Let's practice changing exponential expressions of e to natural logarithms (ln), and vice versa. (Don't use a calculator; it's not necessary. Just follow the definition of the natural logarithm and how it relates to e .)
If e^3^ 20.086, what is x in the expression ln x 3?
“Click here” to check your answer. x = 20.
ln is the inverse of
and stated conversely,
is the inverse of ln
Note: Since ln is the inverse of ,
ln 1
x
x
x
y x y e
y e y x
x e
e
Properties of common logarithms (log) apply to natural logarithms (ln) as well.
Log Rules and Properties (04:34)
Example #2 : Express 3 ln 5 as a single natural logarithm.
3ln 5 ln 5^3 -Power property of logarithms ln125 -Simplify
Properties of Natural Logs
Product property ln ab ln a ln b
Quotient property ln^ a^ ln a ln b
b
Power property ln m p p ln m
Exponential & Logarithmic e ln^ x x
Inverse property ln e x x for x 0
One-to-one property if e x e y , then x y
of exponents
One-to-one property if ln x ln y , then x y
of logarithms
Example #3 : Express ln 35 – ln 5 as a single natural logarithm.
ln 35 ln 5 ln 35 -Quotient property of logarithms 5 ln 7 -Simplify
Example #4 : Express 2 ln x + 3 ln y + ln 8 as a single natural logarithm.
2 3 2 3
2 ln 3ln ln 8
ln ln ln 8 -Power property of logarithms
ln 8 -Product property of logarithms
x y
x y
x y
Stop! Go to Questions #1-8 about this section, then return to continue on to the next section.
Example #3 : Solve 3 4 22
x e for x. 3
3
3
3
18 -Subtract 4 from both sides.
ln ln18 -Take the natural log of both sides.
ln18 -Remember , ln. 3 3 3(ln18) -Multiply both sides by 3.
l
-Use a calculat r
n
o
x
x
x
x
x e
e
e
e x (^) e x
x x
x
Example #4 : Solve 8 e^2 x ^5 56 for x.
8 e^2 x ^5 56 e^2 x ^5 7 -Divide both sides by 8.
ln e^2 x ^5 ln 7 -Take the natural log of both sides.
2 x 5 ln 7 -Remember ln ex x , ln e^2^ x ^5 2 x 5.
2 x ln 7 5 -Add 5 to both sides.
ln 7 5 2 x - Divide by 2.
x 3.47 -Use a calculator.
Example #5 : Solve 500 100 e 0.75 t for t.
500 100 e^ 0.75 t
5 e^ 0.75 t -Divide both sides by 100.
ln 5 ln e 0.75 t -Take the natural log of both sides.
ln 5 0.75 t -Remember ln ex x , ln e 0.75 t 0.75. t
ln 5
t -Divide both sides by 0.75.
t 2.1459 -Use a calculator.
Putting Exponential Equations in Terms of e (04:59)
Example #6 : Solve ln(10 ) x ln(3 x 14)for x.
ln( ) ln( ) 10 3 14 -Apply the One-to-one property. 7 14 -Subtract 3 from both sides. 2 -Divide both sides by
x x x
x
x x
x
Example #7 : Solve ln x ln( x 3) ln10for x.
ln x ln( x 3) ln
ln[ ( x x 3)] ln10 -Apply the Product property.
x x ( 3) 10 -Apply the One-to-one property
x^2 3 x 10 -Distribute
x^2 3 x 10 0 -Subtract 10 from both sides, then apply the zero product property to solve.
( x – 5)( x + 2) = 0 -Factor x – 5 = 0 or x + 2 = 0 -Set both factors equal to zero.
x = 5 or x = –2 -Solve Check both of these answers in the original problem. If either solution results in a negative logarithm, that solution must be denied.
Check : for 5 for – ln x ln( x 3) ln10 ln x ln( x 3) ln
ln 5 ln(5 3) ln10 ln( 2) ln( 2 3) ln
ln 5 ln 2 ln
ln( 2) ln( 5) ln
Applications of Natural Logarithms
A formula using natural logarithms is the continuous compound interest formula where A is the final amount, P is the amount invested, r is the interest rate, and t is time.
Example #1 : Find the value of $500 after 4 years invested at an annual rate of 9% compounded continuously.
A = unknown P = $500 r = 9% = 0.09 t = 4 years
500 (0.09)(4) -Substitute in , , and. $716.66 -Evaluate and round to the nearest cent.
A Pe^ rt A e P r t A
*On the calculator there is a feature for inputting e x. That feature is found by pressing
2 nd^ LN. Notice there is an e x above the LN button. Press 500, then 2nd^ LN, then (0.09 ×
4). Press enter and the answer appears!
A Pe^ rt
A = final amount
P = the initial amount of money invested r = the interest rate (% written as a decimal value)
t = time in years
Example #2 : How long will it take to double your money if you deposit $500 at an annual rate of 7.2% compounded continuously? Consider that the money deposited is $500, the initial amount. This amount doubled is $1000 and that will be the final amount to determine ( A ).
Let t = the amount of time it will take to double the money.
A Pe^ rt
1000 500 e^ (0.072) t -Substitute A = 1000, P = 500, r = 0.
2 e (0.072) t - Divide both sides by 500.
ln 2 ln e (0.072) t -Take the natural log of both sides.
ln 2 0.072 t -Remember l n ex x , ln e (0.072) t 0.072 t. ln 2
t -Divide both sides by 0.072.
t 9.63 -Use the calculator to find the natural log of 2, then divide that answer by 0.072.
Stop! Go to Questions #15-30 to complete this unit.
