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Unit overview In this unit you will evaluate natural exponential and also natural logarithmic functions and model exponential growth and decay processes. You will additionally solve logarithmic and exponential equations by using algebra and also graphs. Real world problems involving exponential and also logarithmic relationships will be resolved at the conclusion of the unit.

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Natural Exponential and also Natural Logarithmic FunctionsThe number e is an important irrational number. The is roughly equal come 2.71828. Like pi, the value is constant.

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one exponential role with the basic e is dubbed a organic base exponential function. These features are advantageous in describing continuous growth or decay. Because that example, the number e is provided to solve problems involving constant compound attention and constant radioactive decay. Exponential features with the base e have the exact same properties as other exponential function.The natural logarithmic function y = log in e x is abbreviation y = ln x and is the train station of the organic exponential role y = e x.
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Example #1: If e5 = 148.413, what is x in the expression ln x = 5? e5 = 148.413
ln(148.413) = 5 Write the exponential as the inverse logarithm (ln).
x = 148.413
*Note: The 3 triangular dots room an abbreviation because that the word, “therefore”. Example #2: If ln 54.598 = 4, what is x in the expression ex = 54.598? ln 54.598 = 4
e4 = 54.598 Write the exponential together the inverse logarithm (ln).
x = 4
Let"s practice transforming exponential expression of e to herbal logarithms (ln), and vice versa. (Don"t usage a calculator; it"s not necessary. Just follow the meaning of the organic logarithm and also how that relates come e.)

If e3 = 20.086, what is x in the expression ln x = 3?


x = 20.086


"Click here" to inspect the answer.

If e−2 = 0.0135, what is x in the expression ln 0.0135= x?


x = −2


"Click here" to check the answer.

If ln 7.389 = 2, what is x in the expression e2 = x?


x = 7.389


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If ln 1 = 0, what is x in the expression e0 = x?


x = 1


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If e = x, what is the value of x?


x = 1


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The Calculator and the herbal Log (LN ) The an essential marked LN ~ above the calculator is the herbal logarithm key.
Example #1: advice the expressions making use of a calculator.
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Properties of common logarithms (log) apply to organic logarithms (ln) together well.
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log Rules and Properties (04:34)
Example #2: express 3 ln 5 as a single natural logarithm. 3 ln 5 = ln 53 -Power residential or commercial property of logarithms
= ln 125 -Simplify
Example #3: express ln 35 – ln 5 as a single natural logarithm. ln 35 – ln 5 = ln
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-Quotient residential or commercial property of logarithms
= ln 7 -Simplify
Example #4: express 2 ln x + 3 ln y + ln 8 as a single natural logarithm. 2 ln x + 3 ln y + ln 8 =
ln x2 + ln y3 + ln 8 = -Power home of logarithms
ln 8x2y3 -Product property of logarithms
Stop! walk to questions #1-8 about this section, climate return to continue on to the next section. resolving Equations with organic Logarithms currently let’s take it a look at using the natural log to deal with an equation.
Example #1: fix 6.5x = 44 for x. ln 6.5x= ln 44 -Take the natural log that both sides.
x ln 6.5 = ln 44 -Apply the strength property.
x =
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-Divide both political parties by ln 6.5.
x ≈ 2.02 -Round the answer to the nearest hundredth.

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To check the above problem, instead of 2.02 in for x in the initial equation.
6.5x = 44
6.52.02 ≈ 43.86 -Use the calculator to evaluate.
*Note: This trouble can likewise be solved by taking the common logarithm (log) that both sides of the equation. natural logarithms (ln) must be used to solve problems that save on computer the number e.
Example #2: deal with ex = 40 because that x. ex = 40
ln ex = ln 40 -Take the herbal log of both sides.
x = ln 40 -Remember ln ex = x.
x ≈ 3.69 -Use a calculator.
Example #3: Solve+ 4 = 22 for x. + 4 = 22
= 18 -Subtract 4 native both sides.
ln = ln 18 -Take the herbal log of both sides.
= ln 18 -Remember ln ex = x, ∴ln = .
x = 3(ln 18) -Multiply both political parties by 3.
x ≈ 8.67 -Use a calculator.
Example #4: Solve8e2x−5 = 56 for x. 8e2x−5 = 56
e2x−5 = 7 -Divide both sides by 8.
ln e2x−5 = ln 7 -Take the organic log of both sides.
2x − 5 = ln 7 -Remember ln ex = x, ∴ln e2x−5 = 2x − 5.
2x = ln 7 + 5 -Add 5 come both sides.
x =
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-Divide by 2.
x ≈ 3.47 -Use a calculator.
Example #5: Solve500 = 100e0.75t for t. 500 = 100e0.75t
5 = e0.75t -Divide both political parties by 100.
ln 5 = ln e0.75t -Take the herbal log that both sides.
ln 5 = 0.75t -Remember ln ex = x, ∴ln e0.75t = 0.75t.
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= t
-Divide both sides by 0.75.
t ≈ 2.1459 -Use a calculator.
Putting Exponential Equations in terms of e (04:59)
Example #6: Solveln (10x) = ln(3x + 14) because that x. ln (10x) = ln(3x + 14)
10x = 3x + 14 -Apply the One-to-one property.
7x = 14 -Subtract 3x indigenous both sides.
x = 2 -Divide both sides by 2.
Example #7: Solveln x + ln (x − 3) = ln 10 for x. ln x + ln (x − 3) = ln 10
ln <x(x − 3)> = ln 10 -Apply the Product property.
x(x − 3) = 10 -Apply the One-to-one property.
x2 − 3x = 10 -Distribute
x2 − 3x − 10 = 0 -Subtract 10 from both sides, then apply the Zero product residential or commercial property to solve.
(x – 5)(x + 2) = 0

-Factor

x – 5 = 0 or x + 2 = 0

-Set both components equal to zero.

x = 5 or x = −2 -Solve
inspect both of these answers in the initial problem. If either solution outcomes in a an unfavorable logarithm, the solution need to be denied.
Check: for 5 for –2
ln x + ln (x − 3) = ln 10 ln x + ln (x − 3) = ln 10
ln 5 + ln (5 − 3) = ln 10 ln (–2) + ln (–2 − 3) = ln 10
ln 5 + ln 2 = ln 10
ln (–2) + ln (–5) = ln 10
2.30 = 2.30
*This equation yields two hopeful logarithms because of this 5 is a solution. *This equation yields two an adverse logarithms as such –2 can not be a a solution since there room no negative logarithms.
Stop! go to inquiries #9-14 about this section, then return to continue on come the next section.Applications of organic Logarithms A formula using herbal logarithms is the continuous compound attention formula wherein A is the final amount, P is the quantity invested, r is the attention rate, and also t is time.
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Example #1: uncover the value of $500 after 4 year invested at an yearly rate the 9% compounded continuously. A = unknown P = $500 r = 9% = 0.09 t = 4 years
A = Pert
A = 500e(0.09)(4) -Substitute in P, r, and t.
A ≈ $716.66 -Evaluate and round to the nearest cent.
*On the calculator there is a attribute for inputting ex. That attribute is found by pressing second LN. Notification there is one exabove the LN button. Push 500, then 2nd LN, then (0.09 × 4). Press enter and also the prize appears!
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Example #2: exactly how long will it take to double your money if you deposit $500 in ~ an annual rate the 7.2% compounded continuously?

Consider the the money deposit is $500, the initial amount. This amount double is $1000 and that will certainly be the final amount to identify (A). allow t = the quantity of time it will take to double the money.

A = Pert
1000 = 500e(0.072)t -Substitute A = 1000, P = 500, r = 0.072
2 = e(0.072)t - divide both sides by 500.
ln 2 = ln e(0.072)t

-Take the herbal log the both sides.

ln 2 = 0.072t -Remember ln ex = x, ∴ln e(0.072)t = 0.072t.
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= t

-Divide both political parties by 0.072.

t ≈ 9.63 -Use the calculator to find the natural log the 2, then divide that answer by 0.072.
Stop! walk to concerns #15-30 to complete this unit.

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below are added educational resources and tasks for this unit.

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click the symbol to find and also practice topics for this unit.
Properties of Logarithms
Inverse the Logarithms