Let be a positive real number different from and let

MATHEMATICS · Subjective · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Let

b

be a positive real number different from

1

and let

f(x)=b^{x}

. (i) Suppose

b>1

. Show that

f

is strictly increasing on

\mathbb R

by comparing the values

f(x_{1})

and

f(x_{2})

for

x_{1}>x_{2}

. (ii) Suppose

0<b<1

. Show that

f

is strictly decreasing on

\mathbb R

by a method analogous to part (i). (iii) Explain, in not more than three sentences, why the two proofs together justify the usual terminology of “exponential growth” for

b>1

and “exponential decay” for

0<b<1

Related theory

Growth Models
Exponential Functions
Exponential and Logarithmic Functions
Functions

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