Cracking the SAT subject test in math 2, 2nd edition

Cracking the SAT Subject Test in Math 2, 2nd Edition LOGARITHMS Exponents can also be written in the form of logarithms For example, log2 8 represents the exponent that turns 2 into 8 In this case, th[.]

Exponents can also be written in the form of logarithms For example, log2 8 represents the exponent that turns 2 into 8 In this case, the “base”

of the logarithm is 2 It’s easy to make a logarithmic expression look like a normal exponential expression Here you can say log2 8 = x, where x is

the unknown exponent that turns 2 into 8 Then you can rewrite the equation as 2x = 8 Notice that, in this equation, 2 is the base of the exponent, just as it was the base of the logarithm Logarithms can be rearranged into exponential form using the following definition:

Definition of a Logarithm

logb n = x ⇔ b x = n

A logarithm that has no written base is assumed to be a base-10 logarithm Base-10 logarithms are called “common logarithms,” and are

so frequently used that the base is often left off Therefore, the expression

“log 1,000” means log10 1,000 Most calculations involving logarithms are done in base-10 logs When you punch a number into your calculator and hit the “log” button, the calculator assumes you’re using a base-10 log There will be times when you’re dealing with other bases

Let’s look at an example:

13 log7 22 =

(A) 0.630 (B) 0.845 (C) 1.342 (D) 1.588

(E) 3.143

Here’s How to Crack It

Your calculator (likely) doesn’t have a button to input a different base into a logarithmic expression, so you’ll need to do a couple of tricks to solve this The expression given is the equivalent of 7x = 22 if you put x in

for what log7 22 is equal to At this point, you could Plug In The Answers

for x and see which answer works in the above equation, but there’s also

another approach

You can take the log (base 10) of both sides of the equation:

log 7x = log 22

As you’ll see in the next pages, the Power Rule lets you take the x in the exponent out of the expression and simply multiply the expression by x:

x log 7 = log 22

Now, to isolate x, you divide both sides by log 7:

The answer is (D)

Rather than going through all these steps each time you have a logarithm question with a weird base, you can simply remember the following formula:

Change of Base Formula

Test your understanding of the definition of a logarithm with the following exercises The answers can be found in Part IV

1 log2 32 = _

2 log3x = 4: x =

3 log 1000 =

4 logb 64 = 3: b =

5 xlogx y =

6 log7 1 =

7 logx x =

8 logx x12 =

9 log 37 = _

10 log 5 =

Logarithmic Rules

In addition to the simple questions you just did, you may need to manipulate equations with logarithms There are three properties of logarithms that are often useful on the SAT Subject Test in Math 2 These properties are very similar to the rules for working with exponents— which isn’t surprising, because logarithms and exponents are the same thing The first two properties deal with the logarithms of products and quotients

The Product Rule

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

The Quotient Rule

These rules are just another way of saying that when you multiply terms, you add exponents, and when you divide terms, you subtract exponents

Be sure to remember that when you use them, the logarithms in these cases all have the same base

The third property of logarithms deals with the logarithms of terms raised to powers

The Power Rule

logb (x r ) = r log b x

This means that whenever you take the logarithm of a term with an exponent, you can pull the exponent out and make it a coefficient

log (72) = 2 log 7 = 2(0.8451) = 1.6902

log3 (x5) = 5 log3x

These logarithm rules are often used in reverse to simplify a string of logarithms into a single logarithm Just as the product and quotient rules can be used to expand a single logarithm into several logarithms, the same rules can be used to consolidate several logarithms that are being added or subtracted into a single logarithm In the same way, the power rule can be used backward to pull a coefficient into a logarithm, as an exponent Take a look at how these rules can be used to simplify a string

of logarithms with the same base

In the following exercises, use the Product, Quotient, and Power rules of logarithms to simplify each logarithmic expression into a single logarithm with a coefficient of 1 The answers can be found in Part IV

1 log 5 + 2 log 6 − log 9 =

2 2 log5 12 − log5 8 − 2 log5 3 =

3 4 log 6 − 4 log 2 − 3 log 3 =

4 log4 320 − log4 20 =

5 2 log 5 + log 3 =

Logarithms in Exponential Equations

Logarithms can be used to solve many equations that would be very difficult or even impossible to solve any other way The trick to using logarithms in solving equations is to convert all of the exponential expressions in the equation to base-10 logarithms, or common logarithms Common logarithms are the numbers programmed into your calculator’s logarithm function Once you express exponential equations

in term of common logarithms, you can run the equation through your calculator and get real numbers

When using logarithms to solve equations, be sure to remember the

meaning of the different numbers in a logarithm Logarithms can be converted into exponential form using the definition of a logarithm provided at the beginning of this section

Let’s take a look at the kinds of tough exponential equations that can be solved using logarithms:

39 If 5x = 2700, then what is the value of x ?

This deceptively simple equation is practically impossible to solve using conventional algebra Two to the 700th power is mind-bogglingly huge;

there’s no way to calculate that number There’s also no way to get x out

of that awkward exponent position This is where logarithms come in Take the logarithm of each side of the equation

log 5x = log 2700

Now use the Power Rule of logarithms to pull the exponents out

x log 5 = 700 log 2

Then isolate x.

Now use your calculator to get decimal values for log 2 and log 5, and plug them into the equation

And voilà, a numerical value for x This is the usual way in which

logarithms will prove useful on the SAT Subject Test in Math 2 Solving tough exponent equations will usually involve taking the common log of both sides of the equation, and using the Power Rule to bring exponents down Another method can be used to find the values of logarithms with

bases other than 10, even though logarithms with other bases aren’t programmed into your calculator For example:

25 What is the value of x if log3 32 = x ?

You can’t do this one in your head The logarithm is asking, “What exponent turns 3 into 32?” Obviously, it’s not an integer You know that the answer will be between 3 and 4, because 33 = 27 and 34 = 81 That might be enough information to eliminate an answer choice or two, but it probably won’t be enough to pick one answer choice Here’s how to get an exact answer:

And there’s the exact value of x.

DRILL 3: LOGARITHMS IN EXPONENTIAL

EQUATIONS

In the following examples, use the techniques you’ve just seen to solve these exponential and logarithmic equations The answers can be found

in Part IV

1 If 24 = 3x , then x =

2 log5 18 =

3 If 10n = 137, then n=

4 log12 6 =

5 If 4x = 5, then 4x + 2 =

6 log2 50 =

7 If 3x = 7, then 3x + 1 =

8 If log3 12 = log4x, then x =

Natural Logarithms

On the SAT Subject Test in Math 2, you may run into a special kind of logarithm called a natural logarithm Natural logarithms are logs with a

base of e, a constant that is approximately equal to 2.718.

The constant e is a little like π It’s a decimal number that goes on forever

without repeating itself, and, like π, it’s a basic feature of the universe Just as π is the ratio of a circle’s circumference to its diameter, no matter

what, e is a basic feature of growth and decay in economics, physics, and

even in biology

The role of e in the mathematics of growth and decay is a little

complicated Don’t worry about that, because you don’t need to know

very much about e for the SAT Subject Test in Math 2 Just memorize a

few rules and you’re ready to go

Natural logarithms are so useful in math and science that there’s a special

notation for expressing them The expression ln x (which is read as “ell-enn x”) means the log of x to the base e, or log e x That means that there

are three different ways to express a natural logarithm

Definitions of a Natural Logarithm

ln n = x ⇔ log e n = x ⇔ e x = n

You can use the definitions of a natural logarithm to solve equations that

contain an e x term Since e equals 2.718281828…, there’s no easy way to

raise it to a specific power By rearranging the equation into a natural

logarithm in “ln x” form, you can make your calculator do the hard work

for you Here’s a simple example:

19 If e x = 6, then x =

(A) 0.45 (B) 0.56 (C) 1.18 (D) 1.79 (E) 2.56

Here’s How to Crack It

The equation in the question, e x = 6, can be converted directly into a logarithmic equation using the definition of a logarithm It would then be written as loge 6 = x, or ln 6 = x To find the value of x, just hit the “LN” key on your calculator and punch in 6 You’ll find that x = 1.791759 The

correct answer is (D)

Calculator Tip

On some scientific calculators, you’ll punch in 6

first, and then hit the “ln x” key

Plugging In, PITA, and Logarithms

Because many logarithm questions involve algebraic manipulation, you can Plug In or Plug In the Answers on most of these questions Typically, this approach is faster than using the logarithmic rules Be sure to remain

flexible in your approach and look for the most efficient way to do each problem

Graphing Logarithmic and Exponential

Functions

For the SAT Subject Test in Math 2, you may also have to know the shapes of some basic graphs associated with natural logs

Here they are:

Finally, some questions may require you to estimate the value of e to answer a question Just remember that e ≈ 2.718 If you forget the value

of e, you can always get your calculator to give it to you Just hit the “2nd” key followed by the “LN” key, and punch in 1 The result will be e to the first power, which is just plain e.

DRILL 4: NATURAL LOGARITHMS

The answers can be found in Part IV

8 If the graph above shows f(x), then which of the

following could be f(x)?

(A) ln x (B) −ln x (C) e x

(D) e −x

(E) −e x

18 If e z = 8, then z =

(A) 1.74

(B) 2.08

(C) 2.35

(D) 2.94

(E) 3.04

23 If set M = {π, e, 3}, then which of the following shows the elements in set M in descending order?

(A) {π, e, 3}

(B) {e, 3, π}

(C) {π, 3, e}

(D) {3, π, e}

(E) {3, e, π}

38 If , then what is the value of n ?

(A) −0.55

(B) −0.18

(C) 0.26

(D) 0.64

(E) 1.19

40 If ln 1.5x = 1.5, then x =

(A) 0

(B) 0.270

(C) 0.405

(D) 2.988

(E) 4.481