AP Precalculus · Unit 2: Exponential and Logarithmic Functions · Lesson 2.4

Deep Dive: Exponential Function Manipulation

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

2.4.A2.4.A.1Concept

The Product Property: Shifts Become Stretches

When you multiply two powers of the same base, you add the exponents:

bm · bn = bm + n

For example, 2³ · 2⁴ = 8 · 16 = 128 = 2⁷. Count the factors: three factors of 2 and four more make seven.

Now apply the property to a horizontal translation of an exponential function. Start with f(x) = b^{x + k}. Split the exponent:

bx + k = bk · bx = a·bx  where  a = bk

So every horizontal translation of an exponential function is equivalent to a vertical dilation. Shifting the graph left or right gives the same graph as stretching or shrinking it up and down.

Translated form	Dilated form	Why
2^{x + 3}	8·2^x	2³ = 8
3^{x − 2}	(1/9)·3^x	3⁻² = 1/9
5·2^{x + 1}	10·2^x	2¹ = 2, and 5·2 = 10

Pick a base b and a shift k. The blue curve is the translated function b^{x + k}. The gold dots come from the dilated form a·b^x with a = b^k. They land on the curve every time, so the two forms are the same function.

shift k3

x	0	1	2	3
2^{x + 3}	8	16	32	64
8·2^x	8	16	32	64

2x + 3 = 23·2x = 8·2xShifting the graph 3 units to the left is the same as dilating it vertically by the factor a = 2^3 = 8.

🔑The dilated form a·b^x shows the initial value right away: a is the output at x = 0. The translated form hides it inside the exponent. That is why rewriting is useful.

💡Check with 2^{x + 3} and 8·2^x at x = 2: 2⁵ = 32 and 8·4 = 32 ✓.

2.4.A2.4.A.2Concept

The Power Property: Stretches Become New Bases

When you raise a power to a power, you multiply the exponents:

(bn)m = bmn

For example, (2³)² = 8² = 64 = 2⁶.

Apply this to a horizontal dilation of an exponential function, f(x) = b^cx, where c is a nonzero constant:

bcx = (bc)x  with new base  bc

So every horizontal dilation of an exponential function is equivalent to a change of the base. Squeezing or stretching the graph sideways gives the same graph as an exponential function with a different base.

Dilated form	New-base form	Why
2^3x	8^x	2³ = 8
4^x/2	2^x	4^1/2 = 2
2^−x	(1/2)^x	2⁻¹ = 1/2

The function b^cx is a horizontal dilation of b^x. Pick b and c. The blue curve is b^cx and the gold dots come from (b^c)^x, a plain exponential function with the new base b^c.

22x = (22)x = 4xCompressing the graph horizontally by the factor 2 makes the base larger: the output is multiplied by 4 for each step of 1.

🔑The base tells you the factor the output is multiplied by for each step of 1 in the input. If the exponent is cx, the new base b^c is the factor per unit input. This is how you change the time unit: P(t) = 500·2^t/3 doubles every 3 days, and the equivalent form 500·(2^1/3)^t ≈ 500(1.26)^t shows it grows by about 26% each day.

⚠️The two properties do different jobs. A translation (x + k) turns into a multiplier in front, a = b^k. A dilation (cx) turns into a different base, b^c. Do not mix them up.

2.4.A2.4.A.3Concept

The Negative Exponent Property

A negative exponent means a reciprocal:

b−n = 1 / bn

For example, 2⁻³ = 1/8 = 0.125, 5⁻² = 1/25 = 0.04, and 10⁻³ = 0.001. A negative exponent does not make the number negative. It tells you to divide instead of multiply.

This agrees with the product property: bⁿ · b⁻ⁿ = b^{n + (−n)} = b⁰ = 1, so b⁻ⁿ must be the reciprocal of bⁿ.

A negative exponent means a reciprocal: b⁻ⁿ = 1 / bⁿ. The blue curve is b^x. The gold curve is (1/b)^x, which equals b^−x. Slide n and compare the points at x = n and x = −n.

2−3 = 1 / 23 = 1 / 8 = 0.125The product 23 · 2−3 = 8 · 0.125 = 1, as the product property says: 23 + (−3) = 20 = 1. The gold curve is the blue curve reflected over the y-axis.

💡Combine it with the power property: (1/b)^x = (b⁻¹)^x = b^−x. So the function 3·2^−x can be written as 3·(1/2)^x. It is an exponential decay function with a = 3 and b = 1/2.

⚠️Common slip: 3⁻² = 1/9, not −9 and not −6. The sign of the exponent controls whether you multiply or divide, never the sign of the result.

2.4.A2.4.A.4Concept

Unit Fraction Exponents Are Roots

An exponent of the form 1/k, where k is a natural number, is a unit fraction. The value of b^1/k is the kth root of b, when it exists:

b1/k = k√b  since  (b1/k)k = b(1/k)·k = b

The check uses the power property: raising b^1/k to the kth power gives b back.

Expression	Value	Check
9^1/2	3	3² = 9
8^1/3	2	2³ = 8
16^1/4	2	2⁴ = 16
5^1/2	≈ 2.236	(2.236)² ≈ 5

An exponential unit fraction, b^1/k, is the kth root of b, when it exists. Pick b and k. The check is that raising the result to the kth power gives back b.

(8)1/3 = the cube root of 8= 2check: (2)3 = 8 ✓

Putting the properties together

Use the power property to build other exponents: 8^2/3 = (8^1/3)² = 2² = 4, and with a negative exponent, 27^−1/3 = 1 / 27^1/3 = 1/3.

🔑Why this matters for models. If a quantity doubles every 3 days, then P(t) = 500·2^t/3 = 500·(2^1/3)^t. The unit fraction exponent gives the daily growth factor 2^1/3 ≈ 1.26. Similarly, 2^d written as (2⁷)^d/7 = 128^d/7 shows the quantity grows by a factor of 128 every week.

⚠️"When it exists" matters. An even root of a negative number, like (−4)^1/2, is not a real number. An odd root of a negative number, like (−8)^1/3 = −2, is. The base of an exponential function is always positive, so its roots always exist.

← Back to Lesson 2.4 Try Guided Practice →Ready for 2.5? Exponential Function Context and Data Modeling puts these equivalent forms to work on real situations.