MA 1128: Lecture 21 – 4/26/11

Natural Exponents and Logs

Exponential and Logarithmic Equations.

Natural Exponents and Logs

In the last lecture, I said that we can get by with just one log function.

It used to be that complicated computations involving multiplication, division, square roots, etc. were done using common logs (log10).

We use calculators for that stuff now, so common logs aren’t really that common anymore.

Today, certain properties of log functions and exponential functions are useful in many applications.

In these contexts, another log function is more convenient.

It is called the natural log.

The natural log is a log with a funny base, a number called e.

The number e is a special constant, kind of like .

The exponential function ex and the natural log function loge(x) turn out to be the “nicest” exponential and log functions for reasons I won’t get into.

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Cont.

The constant e is approximately 2.718281828.

You should remember that it’s around 2.7, or little less than 3.

The natural log is usually written ln rather than loge.

You should have an ln-button on your calculator (maybe “LN”).

There should also be an ex-button.

With the e, things look a little different, but just remember, they’re just like the exponential and log functions we have already seen.

Consider the equation ex = 5. We want to undo the exponential function with the natural log.

So we have, ln(ex) = ln(5). The ln and the “e to the…” undo each other, so

x = ln(5) = 1.609437912.

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We Can Use ln as our only log function.

We had a formula that allowed us to compute any log using just the common log button on our calculators.

The same formula works with the natural log.

For example, log7(5) = ln(5)/ln(7) = 0.827087475.

[[ 5, ln, , 7, ln, =. ]]

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Practice Problems

Round your answers to 4 decimal places.

1. Compute ln(1.72).

2. Compute e5.

3. Solve ex = 10.

4. Solve ex = 1.

Answers:

1) 0.5423

2) 148.4132 (some of you will hit the ex-button before the 5, some after)

3) 2.3026

4) 0

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Exponential and Logarithmic Equations

We’ve solved a number of equations that involved exponential functions.We used the corresponding log function to undo the exponential function.If there is a log in an equation, then we simply apply the corresponding

exponential.Here are more examples.

Consider the equation 32x-1 = 7.

This is 3 raised to something, so take log3 of both sides.

We have log3(32x-1) = log3(7).

Since log3 and 3 raised to the… undo each other,

we have 2x – 1 = log3(7).

Then 2x = log3(7) + 1, and x = (log3(7) + 1)/2 = 1.385621875.[[ 7, ln, , 3, ln, =, + 1, =, 2, =.]]

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More Examples

7.2 = e37x.

ln(7.2) = ln(e37x)

ln(7.2) = 37x

ln(7.2)/37 = x

x = .053353541

log7(x) = 3

7log7(x) = 73

x = 343.

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More Examples

ln(x) = 2.

eln(x) = e2

x = e2 = 7.389056099

log5(x – 2)2 = 2

(x – 2)2 = 52

x – 2 = 5

x = 7, -3.

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Practice Problems

Round your answers to 4 decimal places.

1. 3x+2 = 4.

2. e7x-1 = 3.

3. log4(x + 7) = 1.5.

4. ln(x2 + 1) = 2.

Answers:

1) x = log3(4) – 2 = 0.7381

2) x = [ln(3)+1]/7 = 0.2998

3) x = 41.5 – 7 = 1

4) x = sqrt(e2-1) = 2.5277

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