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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.