Math 140

Class 7

Exponential and Logarithmic Functions

\[y = a^x\]

\[x = \log_a y\]

Basic Properties

Property

Domain:

Range:

Intercept:

Asymptote:

Increase/decrease:

\(f(x) = a^x\)

\(g(y) = \log_a y\)

Graphs

Change of Base Formulas

\[b^x = a^{x\cdot \log_a b}\]

You can get one exponential function from another by horizontal scaling.

\[\log_b x = \frac{\log_a x}{\log_a b}\]

You can get one logarithmic function from another by vertical scaling.

More Properties

Exponents

• \(a^{x+y} = a^x\cdot a^y\)

• \(a^{x - y} = \frac{a^x}{a^y}\)

• \(\left(a^x\right)^r = a^{rx}\)

Logarithms

• \(\log_a (x\cdot y) = \log_a x + \log_a y\)

• \(\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y\)

• \(\log_a\left(x^r\right) = r\log_a x\)

Two Properties

that we have seen before:

If \(f(x) = a^x\) then

• \(f(0) = 1\)

• \(f(x + y) = f(x)\cdot f(y)\)

• \(\operatorname{cis}(0) = 1\)

• \(\operatorname{cis}(x + y) = \operatorname{cis}(x)\cdot \operatorname{cis}(y)\)

Properties of Logarithm

\(\displaystyle \log_5 \frac{50x^3\sqrt{y}}{2z^7} = {}\)

Properties of Logarithm

\(\displaystyle 6\ln x - 3\ln y + \frac{1}{3}\ln z = {}\)

Properties of Logarithm

\(\displaystyle 3 + \log_2 x = {}\)

Scaled Exponential Functions

\[f(x) = b\cdot a^x\]

Transformed Exponential Functions

\[f(x) = b\cdot a^x + c\]

Transformed Logarithmic Functions