Exponential Functions and Equations

SAT Questions that focus on Exponential Functions and Equations require knowledge of the following topics.

Exponential Function is a function of the form

$f(x)=a^x$

where  $a>0$ and $a\neq1$.

For example:
$f(x)=3^x$
$y=(0,5)^x$

Properties of Exponential Functions

1) $a^x.a^y=a^{x+y}$

2) $a^{-x}=\frac{1}{a^x}$

3) $\frac{a^x}{a^y}=a^{x-y}$

4) $(ab)^x=a^x.b^x$

5) $(a^x)^y=a^{x.y}$

6) $\sqrt[x]{a}. \sqrt[x]{b}= \sqrt[x]{a.b}$

7) $\frac{\sqrt[x]{a}} {\sqrt[x]{b}}= \sqrt[x]{\frac{a}{b}}$

8) $\sqrt[y]{\sqrt[x]{a}}= \sqrt[x.y]{a}$

9) $(\sqrt[x]{a})^y= \sqrt[x]{a^y}$

10) $a^{\frac{x}{y}}=\sqrt[y]{a^x}$

Note that some properties can be derived from others. For example, property 3) can de derived from properties 1) and 2):

$\frac{a^x}{a^y}=a^x. \frac {1}{a^y}=a^x.a^{-y}=a^{x-y}$

Graph of Exponential Functions when a>1

For a>1 the graph of the exponential function is monotonically increasing. The larger the value of the base "a", the faster the function grows. The graph below depicts two exponential functions: $f(x)=2^x$ (blue line) and $f(x)=3^x$ (red line).

Graph of Exponential Functions when 0<a<1

For 0<a<1 the graph of the exponential function is monotonically decreasing. The smaller the value of the base "a", the faster the function decreases. The graph below depicts two exponential functions: $f(x)=0.5^x$ (blue line) and $f(x)=0.7^x$ (red line).

Solved SAT Practice Tests

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SAT Practice Tests - Exponential Functions and Equations

Additional Practice Tests - Exponential Functions and Equations