Rewriting the Base, and Why Exponential Always Wins

In this lesson you will: Use (b^m)^n=b^mn to rewrite an exponential expression with a different base; Rewrite a B^kt in the form a C^t to expose the rate per unit of t; Explain and demonstrate why an increasing exponential eventually passes any linear or quadratic function.

One rule does most of the work

A power raised to a power multiplies the exponents: (b^m)^n=b^mn. That single rule is what lets you change the base of an exponential expression.

Write 8^t with base 2. Since 8=2^3, 8^t=(2^3)^t=2^3t. Going the other way, 2^5t=(2^5)^t=32^t.

Nothing has changed about the function — the same inputs give the same outputs. What changes is which feature is easy to see. 32^t shows the factor per unit of t at a glance; 2^5t shows that the underlying step is a doubling that happens five times per unit.

Changing the time unit

An account pays 2% a month. After t years there have been 12t months, so the balance from a $300 deposit is 300 (1.02)^12t.

That is correct but hard to read as a yearly rate. Group the twelve months into one year: 300 (1.02)^12t=300(1.02^12)^t, and 1.02^12 ~ 1.2682.

So the same account grows by about 26.8% a year. The monthly form and the yearly form are the same function; the second one answers \"how much per year\" without any further work.

2 to the x is not x squared

These two expressions get confused constantly, and they are not close relatives. In x^2 the variable is the base and the exponent is fixed. In 2^x the base is fixed and the variable is the exponent.

At small inputs they trade places. At x=2 both are 4; at x=3 the exponential gives 8 while the quadratic gives 9, so the parabola is briefly ahead; at x=4 both are 16; from x=5 onward the exponential is in front and never gives the lead back.

That last sentence is F.LE.3 in miniature. A quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or by any fixed power at all.

Why the exponential has to win

Compare what each function does when x goes up by 1. A linear function adds a fixed amount. A quadratic adds an amount that itself grows, but only by a fixed step. An exponential multiplies.

Multiplying by 2 adds 100% of whatever is already there, so the size of each jump is proportional to the size the function has already reached. Once an exponential is large, its jumps are large, and that feeds back on itself. A polynomial has no such feedback.

Making the linear or quadratic function bigger does not fix this. It only postpones the crossing point.

Common mistake — Adding the exponents instead of multiplying them

Asked to rewrite 8^t with base 2, a student writes 2^3+t.

It comes from mixing up two rules that live next door to each other. b^m* b^n=b^m+n really does add, and (b^m)^n=b^mn really does multiply, and under time pressure the wrong one comes out.

But 8^t is 2^3 multiplied by itself t times, which is 3 twos multiplied t times over — that is 3t twos, not 3+t of them.

The same confusion turns 9^t into 3^2+t and 4^3t into 64+t-shaped nonsense. One rule, applied in the wrong direction, spoils the whole rewrite.

How to catch it: Put t=2 into both forms. The original gives 8^2=64. The correct rewrite gives 2^3* 2=2^6=64. The additive version gives 2^3+2=2^5=32, which is half as big. Any single value of t other than 1 will expose the error — and t=1 is the one value that will not, since both forms give 8.