$\newenvironment {prompt}{}{} \newcommand {\ungraded }{} \newcommand {\todo }{} \newcommand {\mooculus }{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\npnoround }{\nprounddigits {-1}} \newcommand {\npnoroundexp }{\nproundexpdigits {-1}} \newcommand {\npunitcommand }{\ensuremath {\mathrm {#1}}} \newcommand {\tkzTabSlope }{\foreach \x /\y /\z in {#1}{\node [left = 3pt] at (Z\x 1) {\scriptsize \y }; \node [right = 3pt] at (Z\x 1) {\scriptsize \z }; }} \newcommand {\RR }{\mathbb R} \newcommand {\R }{\mathbb R} \newcommand {\N }{\mathbb N} \newcommand {\Z }{\mathbb Z} \newcommand {\sagemath }{\textsf {SageMath}} \newcommand {\d }{\mathop {}\!d} \newcommand {\l }{\ell } \newcommand {\ddx }{\frac {d}{\d x}} \newcommand {\ddt }{\frac {d}{\d t}} \newcommand {\zeroOverZero }{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }{\textbf } \newcommand {\unit }{\mathop {}\!\mathrm } \newcommand {\eval }{\bigg [ #1 \bigg ]} \newcommand {\seq }{\left ( #1 \right )} \newcommand {\epsilon }{\varepsilon } \newcommand {\phi }{\varphi } \newcommand {\iff }{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{\vec {proj}} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }{\overrightarrow } \newcommand {\vec }{\mathbf } \newcommand {\veci }{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }{\boldsymbol {\l }} \newcommand {\uvec }{\mathbf {\hat {#1}}} \newcommand {\utan }{\mathbf {\hat {t}}} \newcommand {\unormal }{\mathbf {\hat {n}}} \newcommand {\ubinormal }{\mathbf {\hat {b}}} \newcommand {\dotp }{\bullet } \newcommand {\cross }{\boldsymbol \times } \newcommand {\grad }{\boldsymbol \nabla } \newcommand {\divergence }{\grad \dotp } \newcommand {\curl }{\grad \cross } \newcommand {\lto }{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }{\overline } \newcommand {\surfaceColor }{violet} \newcommand {\surfaceColorTwo }{redyellow} \newcommand {\sliceColor }{greenyellow} \newcommand {\vector }{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }{} \newcommand {\HyperFirstAtBeginDocument }{\AtBeginDocument }$

Exponential and logarithmic functions illuminated.

Exponential and logarithmic functions may seem somewhat esoteric at first, but they model many phenomena in the real-world.

### What are exponential and logarithmic functions?

Is $b^{-x}$ an exponential function?
yes no

In either definition above $b$ is called the base.

Remember that with exponential and logarithmic functions, there is one very special base: This is an irrational number that you will see frequently. The exponential with base $e$, $f(x) = e^x$ is often called the ‘natural exponential’ function. For the logarithm with base $e$, we have a special notation, $\ln x$ is ‘natural logarithm’ function. We’ll talk about where $e$ comes from when we talk about derivatives.

#### Connections between exponential functions and logarithms

Let $b$ be a positive real number with $b\ne 1$.

• $b^{\log _b(x)} = x$ for all positive $x$
• $\log _b(b^x) = x$ for all real $x$
What exponent makes the following expression true?

### Properties of exponential functions and logarithms

Working with exponential and logarithmic functions is often simplified by applying properties of these functions. These properties will make appearances throughout our work.

#### Properties of exponents

Let $b$ be a positive real number with $b\ne 1$.

• $b^m\cdot b^n = b^{m+n}$
• $b^{-1} = \frac {1}{b}$
• $\left (b^m\right )^n = b^{mn}$
What exponent makes the following true?

#### Properties of logarithms

Let $b$ be a positive real number with $b\ne 1$.

• $\log _b(m\cdot n) = \log _b(m) + \log _b(n)$
• $\log _b(m^n) = n\cdot \log _b(m)$
• $\log _b\left (\frac {1}{m}\right ) = \log _b(m^{-1}) = -\log _b(m)$
• $\log _a(m) = \frac {\log _b(m)}{\log _b(a)}$
What value makes the following expression true?
What makes the following expression true?

### Exponential equations

Let’s look into solving equations involving these functions. We’ll start with a straightforward example.

Of course, if we couldn’t rewrite both sides with the same base, we can still use the properties of logarithms to solve.

Solve the equation: $\displaystyle 2\left (5^{2x} + 6\right ) = 11 \cdot 5^x$.
$\displaystyle \log _{5}\left (\dfrac {3}{2} \right )$ $\displaystyle \frac {\ln \left (\dfrac {3}{2} \right )}{5}$ $\displaystyle \log _{4}\left (5\right )$ $\displaystyle \log _{5}\left (4 \right )$ The equation has no solutions.