Further Exponentials

Meaning and origins

Exponential expressions are expressions that contain an exponent or index.

The exponent, say $m$m, is a small superscripted number placed to the right of a base number $b$b as in $b^m$bm. The original  idea of using exponents was to make the writing down of certain expressions more efficiently.

This means for example that $b\times b=bb=b^2$b×b=bb=b2 and more tedious expressions like $b\times b\times b\times b\times b\times b\times b$b×b×b×b×b×b×b simplified to  $b^7$b7. 

Mathematicians are always inventing notations and symbols that aim to simplify expressions. Their meaning eventually becomes universally accepted and understood.

Sometimes however, we can make mistakes simply because we lose sight of these meanings.

As a simple example of this, the expression $\frac{b^6}{b^2}$b6b2​ is sometimes erroneously simplified by students to $b^3$b3. In fact, if written fully as $\frac{bbbbbb}{bb}$bbbbbbbb​, we see at once that after cancellation, the expression is really $b^4$b4.

As time went by, the nature of the exponents became more generalised, so that $x^{\frac{1}{2}}=\sqrt{x}$x12​=√x, $x^{-2}=\frac{1}{x^2}$x−2=1x2​, $x^{\frac{3}{4}}=\sqrt[4]{x^3}$x34​=⁴√x3 and strange expressions like $x^{\sqrt{2}}$x√2 became more common place in the mathematical literature.  

 

Finding function values

We are often asked to determine functional values of the form $y=a\left(b^x\right)$y=a(bx) or $f\left(x\right)=a\left(b^x\right)$f(x)=a(bx). So for example we can collect function values for the function given by $y=5\left(2^x\right)$y=5(2x) for $x=0,1,2,3,4$x=0,1,2,3,4 by simple substitution. So for $x=0$x=0, $y=5\times2^0=5\times1=5$y=5×20=5×1=5 and for $x=1$x=1, $y=5\times2^1=10\times1=10$y=5×21=10×1=10. Further substitutions complete this table of values:

$x$x	$0$0	$1$1	$2$2	$3$3	$4$4
$y=5\left(2^x\right)$y=5(2x)	$5$5	$10$10	$20$20	$40$40	$80$80

 

 

An Application

When rare paintings are bought, they usually increase in value. They then become what is generally termed an investment.

Suppose a rare Rembrandt painting was purchased for $\$20000$$20000 in 2016 and the value increased by $12%$12% each year for the next $10$10 years.

After one year, the value would become $\left(100+12\right)%$(100+12)% of its purchase price. This means that its value $V_1$V1​ becomes $V_1=20000\times1.12$V1​=20000×1.12 or $\$22400$$22400.

Another year will increase $\$22400$$22400 by the factor $1.12$1.12, so that $V_2=20000\times1.12\times1.12$V2​=20000×1.12×1.12 or $\$25312$$25312. Written more concisely, we can say that $V_2=20000\times1.12^2$V2​=20000×1.122. 

After $3$3 years, $V_3=20000\times1.12^3=$V3​=20000×1.123= $\$28098.56$$28098.56 and after $10$10 years, $V_{10}=20000\times1.12^{10}\approx$V10​=20000×1.1210≈ $\$62117$$62117.

In general, we can say that investments grow in value exponentially, so that an investment bought originally for $\$A$$A, which increases by $r$r $%$% every year for $n$n years will be worth $V_n$Vn​ where:

$V_n=A\left(1+\frac{r}{100}\right)^n$Vn​=A(1+r100​)n

This concisely expressed formula is known as the compound interest formula and has become an extremely important formula in financial circles.

Worked Examples

Question 1

Question 2

Question 3