Exponents & Logarithms

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Exponential Algebra

Discussion

Observing the diagram above, can you tell how many dots come next in the pattern? How do you know?

Many mathematical patterns follow the same basic structure as the dot diagram above: repetitive multiplication by a given number. When we multiply several times by the same number, we use exponential notation.

Exponential Notation

Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:

an=1×a×a×⋯×an times🚫

Here a is called the base and n the exponent. We say that a is raised to the nth power.

Note that a1=a, since we have 1×a=a.

Checkpoint

What is 2.5⋅2.5⋅2.5⋅2.5⋅2.5⋅2.5⋅2.5 in exponential form?

Select the correct option

Discussion

Now, we can interpret the meaning of 2n for any whole number n, but what if n=0? One way to understand this is by finding a pattern in the exponents of 2.

Utilizing the graph above, label each set of dots with an exponential expression of the form 2n. Is 20 shown?

The four clusters have 1, 2, 4 and 8 dots. In expanded form:

8=1⋅2⋅2⋅2=234=1⋅2⋅2=222=1⋅2=21

You may notice that the powers are reducing by 1 as we go to the left and recognize that

1=1⋅( zero 2’s)=20

Or, algebraically, each time we decrease the exponent by 1 we divide by 2:

223=22,222=21,221=20

Hence the clusters are labelled

1=20,2=21,4=22,8=23

In conclusion, yes, 20 is shown: it is the single dot.

Exponent with zero

Any number raised to the power zero is

a0=1×a×a×⋯×a0 times=1🚫

This also applies, somewhat confusingly, when a=0:

00=1×0×0×⋯×00 times=1🚫

Checkpoint

Evaluate 00⋅50.

Select the correct option

Discussion

Considering the 5 clusters above, how many dots would the seventh cluster contain? How do you know?

In the example above, the fastest way to obtain an answer was to continue the doubling pattern represented in the diagram and solving

24⋅2⋅2=24⋅22=16⋅4=64.

But also, if we continued the pattern with labels, you might notice that the seventh cluster would be labeled 26, which also equals 64.

One thing you may have noticed is that 24⋅22=26. In general, when we multiply exponentials with the same base, we can add the powers of the two exponentials. We can see this algebraically:

24⋅22=(2⋅2⋅2⋅2)⋅(2⋅2)=2⋅2⋅2⋅2⋅2⋅2=26.

And this is not unique to the numbers we chose:

53⋅52=(5⋅5⋅5)⋅(5⋅5)=5⋅5⋅5⋅5⋅5=55.

Multiplying exponents

When multiplying exponentials with the same base, the following rule applies:

an⋅am =a×a×⋯×an times×a×a×⋯×am times =a×a×⋯×an+m times =am+n🚫

Checkpoint

It is given that 33⋅37=3n. Find n.

Select the correct option

Discussion

Determine how many dots should be clustered where the question mark is in the diagram above. Explain which pattern(s) support your answer.

Looking at the diagram, we see that each position in the bottom row contains a square of dots. The side length of each square matches the number of dots in the shape directly above it in the top row:

In the first column, the top row has 1 dot, and the bottom row has a 1×1 square (1 dot).
In the second column, the top row has 2 dots, and the bottom row has a 2×2 square (4 dots).
In the third column, the top row has 4 dots, and the bottom row has a 4×4 square (16 dots).

So, the pattern is: The bottom row has a square of dots, with side length equal to the number of dots in the top row.

For the fourth column, the top row has 8 dots. So, the bottom row should have an 8×8 square of dots.

Calculating the total number of dots:

8×8=64

Therefore, the question mark should be replaced with 64 dots.

Otherwise, you might realize that the bottom row of dots is quadrupling between columns. Then, you can identify that the square before the question mark contains 16 dots, so the missing square would have 4⋅16, or 64 dots. Both solutions are equally valid.

One thing you may have noticed about the previous diagram is that the bottom row clusters contain the square of the number of dots in the cluster directly above. Thus, we could label the diagram as:

However, the bottom row clusters have 1, 4, 16, and 64 dots sequentially. That is, in base 2 exponentials, 20, 22, 24, and 26. Therefore, we see:

(20)2=(21)2=(22)2=(23)2=20222426.

You may notice the pattern that the exponential of an exponential simplifies to a single exponential to the power of the product of the original powers.

Here is another example highlighting the algebra behind this pattern:

(32)3=(3⋅3)⋅(3⋅3)⋅(3⋅3)=3⋅3⋅3⋅3⋅3⋅3=36.

Exponential of exponential

An exponential can be the base of another exponential:

(am)n=a×⋯×am times×⋯×a×⋯×am timesn times=anm🚫

Discussion

Noah claims that a(x2)=(ax)2. Do you agree with him? Provide a suitable example or counterexample to support your answer.

Checkpoint

Simplify (35)4.

Select the correct option

Discussion

Exponentials can also have negative powers! Using your knowledge that an⋅am=an+m, can you determine the value of 2−3?

Negative exponents

Our reasoning from the prior discussion can be generalized for any base a and any power n:

an⋅a−n=an−n=a0=1

Isolating a−n we find

a−n=an1🚫

Checkpoint

Evaluate 3−4.

Select the correct option

Discussion

Now, using our understanding of negative exponents, we can figure out how to divide exponentials.

Simplify 7278to a single base 7 exponential. Explain your reasoning.

Dividing exponentials

In general,

aman=an⋅a−m=an−m🚫

Discussion

We already learned that we can combine exponentials with the same bases: an⋅am=an+m, but there is also a way to combine exponentials with the same powers.

How do you think 24⋅54 simplifies?

Exponents of products & quotients

When exponentials with the same power are being multiplied or divided, the bases can be combined:

anbn=(ab)n🚫

bnan=(ba)n🚫

Checkpoint

Simplify 2545⋅65 to a single exponential.

Select the correct option

Exercise

Simplify 83⋅92124.

Select the correct option

Discussion

Considering the equation 42⋅43=4−3⋅4n, solve for n.

The equation you solved above is an example of an exponential equation, and your intuition utilized the strategy of equating indices!

Exponential Equations (Equating Indices)

An exponential equation is an equation which contains a number to the power of a variable expression (i.e. 25⋅22=22x).

Equating indices is a method for solving exponential equations in which we solve an equation ax=ay by solving x=y.

Exercise

Solve the equation 24x+1+20=33.

Select the correct option

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