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It’s the Law Too — the Laws of Logarithms

revised 6 Feb 2012

Copyright © 2002–2012 by Stan Brown, Oak Road Systems

Do you have trouble remembering the laws of logarithms? Do you know why you can

change log(x)+log(y) to a different form, but not log(x+y)? This page helps you make

sense out of the laws of logarithms.

All the laws of logarithms "low directly out of the laws of exponents. If you feel a bit

unsteady with the laws of exponents, please review them before going on.

Logarithm? What’s a Logarithm?

Where Did Logs Come From?

Why Do We Care?

“Base”ic Facts

Log of 1, Log Equaling 1

Log as Inverse

What’s “ln”?

Combining Logs with the Same Base

Multiply Numbers, Add Their Logarithms

Exponent, Multiply the Logarithm

Raising Numbers to Any Power

Divide Numbers, Subtract Their Logarithms

Changing the Base

Summary

Conclusion

What’s New

You’re welcome to print copies of this page for your own use, and to link from your own

Web pages to this page. But please don’t make any electronic copies and publish them on

your Web page or elsewhere.

Logarithm? What’s a Logarithm?

A logarithm is just an exponent.

To be speci"ic, the logarithm of a number x to a base b is just the exponent you put onto b to make

the result equal x. For instance, since 5² = 25, we know that 2 (the power) is the logarithm of 25 to base

5. Symbolically, log (25) = 2.

More generically, if x = b , then we say that y is “the logarithm of x to the base b” or “the base‐b5

y

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logarithm of x”. In symbols, y = log (x). Every exponential equation can be rewritten as a logarithmic

equation, and vice versa, just by interchanging the x and y in this way.

Another way to look at it is that the log x function is de"ined as the inverse of the b function. These

two statements express that inverse relationship, showing how an exponential equation is equivalent

to a logarithmic equation:

x = b is the same as y = log x

Example 1: 1000 = 10 is the same as 3 = log 1000.

Example 2: log 81 = ? is the same as 3 = 81.

It can’t be said too often: a logarithm is nothing more than an exponent. You can write the above

de"inition compactly, and show the log as an exponent, by substituting the second equation into the "irst

to eliminate y:

Read that as “the logarithm of x in base b is the exponent you put on b to get x as a result.”

Where Did Logs Come From?

Before pocket calculators—only three decades ago, but in “student years” that’s the age of

dinosaurs—the answer was simple. You needed logs to compute most powers and roots with fair

accuracy; even multiplying and dividing most numbers were easier with logs. Every decent algebra

books had pages and pages of log tables at the back.

The invention of logs in the early 1600s fueled the scienti"ic revolution. Back then scientists,

astronomers especially, used to spend huge amounts of time crunching numbers on paper. By cutting

the time they spent doing arithmetic, logarithms effectively gave them a longer productive life. The

slide rule, once almost a cartoon trademark of a scientist, was nothing more than a device built for

doing various computations quickly, using logarithms. See Eli Maor’s e: The Story of a Number for more

on this.

Today, logs are no longer used in routine number crunching. But there are still good reasons for

studying them.

Why Do We Care?

Why do we use logarithms, anyway? I could write a whole article about them—maybe one day. But for

now. ...

To "ind the number of payments on a loan or the time to reach an investment goal.

To model many natural processes, particularly in living systems. We perceive loudness of

sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale.

We also perceive brightness of light as the logarithm of the actual light energy, and star

magnitudes are measured on a logarithmic scale.

b

bx

yb

310

3?

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To measure the pH or acidity of a chemical solution. The pH is the negative logarithm of the

concentration of free hydrogen ions.

To measure earthquake intensity on the Richter scale.

To analyze exponential processes. Because the log function is the inverse of the exponential

function, we often analyze an exponential curve by means of logarithms. Plotting a set of

measured points on “log‐log” or “semi‐log” paper can reveal such relationships easily.

Applications include cooling of a dead body, growth of bacteria, and decay of a radioactive

isotopes. The spread of an epidemic in a population often follows a modi"ied logarithmic

curve called a “logistic”.

To solve some forms of area problems in calculus. (The area under the curve 1/x, between x=1

and x=A, equals ln A.)

Also in calculus, differentiating a complicated product becomes much easier if you "irst take

the logarithm.

(Historically, the main reason for teaching logs in grade school was to simplify computation, because

the log of a multiplication “downgrades” it to an addition, and the log of a power expression

“downgrades” it to a multiplication. Of course, with the widespread availability of personal computing

devices, dif"iculty of computation is no longer a concern, but logs still have many applications in their

own right.)

“Base”ic Facts

From the de"inition of a log as inverse of an exponential, you can immediately get some basic facts. For

instance, if you graph y=10 (or the exponential with any other positive base), you see that its range is

positive reals; therefore the domain of y=log x (to any base) is the positive reals. In other words, you

can’t take log 0 or log of a negative number.

(Actually, if you’re willing to go outside the reals, you can take the log of a negative number. The

technique is taught in many trigonometry courses.)

Log of 1, Log Equaling 1

You know that anything to the zero power is 1: b = 1. Change that to logarithmic form with the

de"inition of logs and you have

log 1 = 0 for any base b

In the same way, you know that the "irst power of any number is just that number: b = b. Again, turn

that around to logarithmic form and you have

log b = 1 for any base b

Example 3: ln 1 = 0

x

0

b1

b

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Example 4: log 5 = 1

Log as Inverse

A log is an exponent because the log function is the inverse of the exponential function. The inverse

function undoes the effect of the original function. (I’m not a big fan of most uses the term “cancel” in

math, but it does "it in this situation.)

This means that if you take the log of an exponential (to the same base, of course), you get back to

where you started:

log b = x for any base b

This fact lets you evaluate many logarithms without a calculator.

Example 5: log 125 = log (5³) = 3

Example 6: log 10 = 3.16

Example 7: ln e = -kt/2

What’s “ln”?

Any positive number is suitable as the base of logarithms, but two bases are used more than any others:

base of

logarithmssymbol name

10log

(if no base shown)common logarithm

e lnnatural logarithm,

pronounced “ell‐enn” or “lahn”

Natural logs are logs, and follow all the same rules as any other logarithm. Just remember

ln x means log x

Why base e? What’s so special about e? Most of the explanations need some calculus, for instance that e

is the only function that is both its own integral and its own derivative or that e has this beautiful

de"inition in terms of factorials:

e = 1/0! + 1/1! + 1/2! + 1/3! + ...

Numerically, e is about 2.7182818284. It’s irrational (the decimal expansion never ends and never

repeats), and in fact like pi it’s transcendental (no polynomial equation with integer coef"icients has pi

or e as a root.)

e (like pi) crops up in all sorts of unlikely places, like computations of compound interest. It would

take a book to explain, and fortunately there is a book, Eli Maor’s e: The Story of a Number. He also goes

5

bx

5 5

103.16

-kt/2

e

x

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into the history of logarithms, and the book is well worth getting from your library.

Combining Logs with the Same Base

In a minute we’ll look at the various combinations. But "irst you might want to know the general

principle: logs reduce operations by one level. Logs turn a multiplication into an addition, a division into

a subtraction, an exponent into a multiplication, and a radical into a division. Now let’s see why, and

look at some examples.

Multiply Numbers, Add Their Logarithms

Multiplying two expressions corresponds to adding their logarithms. Can we make sense of this?

By the compact de"inition,

x = b and y = band therefore, substituting for x and y,

xy = b bBut when you multiply two powers of the same base, you add their exponents. So the right‐hand side

becomes

xy = bNow apply the compact de"inition to the left=hand side:

b = xyCombine that with the preceding equation to obtain

b = bNow we have two powers of the same base. If the powers are equal, then the exponents must also be

equal. Therefore

log (xy) = log x + log y

So what’s the bottom line? Multiplying two numbers and taking the log is the same as taking their logs

and adding.

Example 8: log (x)+log (x²) is the same as log (xx²) or just log (x³).

Example 9: log (20)+log (50) = log (20×50) = log (1000) = 3.

Exponent, Multiply the Logarithm

Continuing our theme of logarithms reducing the level of operations, if you have the yth power of a

number and take the log, the result is y times the log of the number. Here’s why, starting with x :

log xb log yb

log xb log yb

log x+log yb b

log (xy)b

log (xy)b log x+log yb b

b b b

8 8 8 8

10 10 10 10

y

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Start with the compact de"inition of a logarithm:

x = band raise both sides to the y power:

x = (b )A power of a power is equivalent to just multiplying the exponents. Simplify the right‐hand side:

x = bRewrite the left‐hand side using the compact de"inition of a log:

b = x(The font may be hard to read: that’s x to the power y on left and right.) and combine the last two

equations:

b = bIf the powers are equal and the bases are equal, the exponents must be equal:

log (x ) = y log x

Example 10: ln(2 ) = 6 ln 2 (where “ln” means log , the natural logarithm).

Example 11: log (5x²) is not equal to 2 log (5x). Be careful with order of operations! 5x² is 5(x²), not

(5x)². log (5x²) must "irst be decomposed as the log of the product: log 5 + log (x²). Then the second

term can use the power rule, log (x²) = 2 log x. The "irst term is just 1. Summing up, log (5x²) = 1 +

2 log x.

Raising Numbers to Any Power

The trick to evaluating expressions like 6.7 is to use the exponent rule and the log‐as‐inverse

de"inition:

x = 6.7

log x = 4.4 ( log 6.7 ) = about 3.634729132

x = 10 = about 4312.5

There’s nothing special about base‐10 logs here. The calculation could just as well be

x = 6.7

ln x = 4.4 ( ln 6.7 ) = about 8.369273116

x = e = about 4312.5

This will work for any positive base and any real exponent, so for example

x = π

log x = π (log π) = about 1.561842388

x = 10 = about 36.46215961

You can combine this with the multiplying numbers = adding logarithms rule to evaluate powers that

are too big for your calculator. For example, what is 671 ?

x = 671

log x = 217 (log 671) = about 613.3987869

Now, separate the integer and fractional parts of the logarithm.

log xb

y log xb y

y (y log x)b

log (x )by y

log (x )by (y log x)b

by

b

6e

5 5

5 5 5

5 5 5

5

4.4

4.4

3.63472...

4.4

8.36927...

π

1.5618...

217

217

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log x = about 0.3987869 + 613

x = 10

x = 10 · 10

x = about 2.505 · 10

For examples like this, you really do have to use base‐10 logs.

If the base is negative or the exponent is complex, see Powers and Roots of a Complex Number.

Divide Numbers, Subtract Their Logarithms

Since division is the opposite of multiplication, and subtraction is the opposite of addition, it’s not

surprising that dividing two expressions corresponds to subtracting their logs. While we could go back

again to the compact de"inition, it’s probably easier to use the two preceding properties.

Start with the fact that 1/y = y (see the de"inition of negative exponents):

x/y = x(1/y) = xyand take the log of both sides:

log (x/y) = log (xy )The right‐hand side is the log of a product, which becomes the sum of the logs:

log (x/y) = log x + log (y )and the second term is the log of a power, which becomes (−1) times the log, or just minus the log:

log (x/y) = log x − log y

In words, if you divide and take the log, that’s the same as subtracting the individual logs.

Example 12: 675÷15=45, and therefore log 675 − log 15 = log 45. (Try it on your calculator!)

Example 13: log(x³y²) − log(x²y³) = log(x³y² / x²y³) = log(x/y) = log(x) − log(y).

Changing the Base

Now you have everything you need to change logarithms from one base to another. Look again at the

compact equation that de"ines a log in base b:

To change the log from base b to another base (call it a), you want to "ind log (x). Since you already

have x on one side of the above equation, it seems like a good start is to take the base‐a log of both

sides:

log (b ) = log x

0.3987869 + 613

0.3987869 613

613

−1

−1

b b−1

b b b−1

b b b

10 10 10

a

alog xb

a

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But the left‐hand side of that equation is just the log of a power. You remember that log(x ) is just log(x)

times y. So the equation simpli"ies to

(log b) (log x) = log x

Notice that log b is a constant. This means that the logs of all numbers in a given base a are

proportional to the logs of the same numbers in another base b, and the proportionality constant log b

is the log of one base in the other base. If you’re like me, you may have trouble remembering whether

to multiply or divide. If so, just derive the equation—as you see, it takes only two steps.

Some textbooks present the change‐of‐base formula as a fraction. To get the fraction from the above

equation, simply divide by the proportionality constant log b:

log x = (log x) / (log b)

Example 14: log 16 = (log 16) / (log 4). (You can verify this with your calculator, since you know

log 16 must equal 2.)

Example 15: Most calculators can’t graph y = log x directly. But you can change the base to e and easily

plot y = (ln x)÷(ln 3). (You could equally well use base 10.)

An interesting side road leads from the above formula. Replace x everywhere with a—this is legal since

the formula is true for all positive a, b, and x. You get

log a = (log a) / (log b)But log a = 1 (see Log of 1 above), so the formula becomes

log a = 1 / (log b)

Example 16: log e = 1/(ln 10). (You can verify this with your calculator.)

Example 17: log 5 = 1/(log 125). This is easy to verify: 5 = 125, and 5 is the cube root of 125.

Therefore log 5 = 1/3 and log 125 = 3, and 1/3 does indeed equal 1/3.

Summary

The laws of logarithms have been scattered through this longish page, so it might be helpful to collect

them in one place. To make this even more amazingly helpful <grin>, the associated laws of exponents

are shown here too.

For heaven’s sake, don’t try to memorize this table! Just use it to jog your memory as needed. Better

yet, since a log is an exponent, use the laws of exponents to re‐derive any property of logarithms that

you may have forgotten. That way you’ll truly gain mastery of this material, and you’ll feel con"ident

about the operations.

exponents logarithms

y

a b a

a

a

a

b a a

4

4

3

b a a

a

b a

10

125 53

125 5

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See also:

(All laws apply for any positive a, b, x, and y.)

x = b is the same as y = log x

b = 1 log 1 = 0

b = b log b = 1

b = x log b = x

b b = b log (xy) = log x + log y

b ÷b = b log (x/y) = log x − log y

(b ) = b log (x ) = y log x

(log b) (log x) = log x

log x = (log x) / (log b)

log a = 1 / (log b)

Don’t get creative! Most variations on the above are not valid.

Example 18: log (5+x) is not the same as log 5 + log x. As you know, log 5 + log x = log(5x), not log(5+x).

Look carefully at the above table and you’ll see that there’s nothing you can do to split up log(x+y) or

log(x−y).

Example 19: (log x) / (log y) is not the same as log(x/y). In fact, when you divide two logs to the same

base, you’re working the change‐of‐base formula backward. Though it’s not often useful, (log x) /

(log y) = log x. Just don’t write log(x/y)!

Example 20: (log 5)(log x) is not the same as log(5x). You know that log(5x) is log 5 + log x. There’s

really not much you can do with the product of two logs when they have the same base.

Combining Operations (Distributive Laws)

Conclusion

Well, there you have it: the laws of logarithms demysti"ied! The general rule is that logs simply drop an

operation down one level: exponents become multipliers, divisions become subtractions, and so on. If

yb

0b

1b

(log x)bb

x

x y x+yb b b

x y x−yb b b

x y xyb

yb

a b a

b a a

b a

y

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ever you’re unsure of an operation, like how to change base, work it out by using the de"inition of a log

and applying the laws of exponents, and you won’t go wrong.

What’s New

6 Feb 2012: Add applications here and here.

6 Mar 2011: Add Raising Numbers to Any Power

(intervening changes suppressed)

11 Jan 1998: Adapt this article for the Web

22 Dec 1997: Post to alt.algebra.help

this page: http://oakroadsystems.com/math/loglaws.htm

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