Exponential and Logarithmic Functions: Learn It 2

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Exponential and Logarithmic Functions: Learn It 2

Exponential Functions Cont.

Evaluating Exponential Functions

To evaluate an exponential function with the form $f(x)=b^x$ , we simply substitute $x$ with the given value, and calculate the resulting power.

Let $f(x)=2^x$ . What is $f(3)$ ?

\begin{array}{rcl} f (x) & = & 2^{x} \\ f (3) & = & 2^{3} & Substitute x = 3. \\ = & 8 & Evaluate the power. \end{array}

$\begin{array}{rcl} f(x) & = & 2^x \\ f(3) & = & 2^3 & \quad \text{Substitute } x = 3. \\ & = & 8 & \quad \text{Evaluate the power.} \end{array}$

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations.

Let $f(x)=30(2)^x$ . What is $f(3)$ ?

\begin{array}{rcll} f (x) & = & 30 (2)^{x} \\ f (3) & = & 30 (2)^{3} & Substitute x = 3. \\ = & 30 (8) & Simplify the power first. \\ = & 240 & Multiply. \end{array}

$\begin{array}{rcll} f(x) & = & 30(2)^x & \\ f(3) & = & 30(2)^3 & \quad \text{Substitute } x = 3. \\ & = & 30(8) & \quad \text{Simplify the power first.} \\ & = & 240 & \quad \text{Multiply.} \end{array}$

Note that if the order of operations were not followed, the result would be incorrect:

f (3) = 30 (2)^{3} \neq 60^{3} = 216, 000

$f(3)=30(2)^3≠60^3=216,000$

How To: Evaluating Exponential Functions

Given an exponential function, identify $a$ , $b$ , and the value of $x$ you’re being asked to substitute into the function.
Replace the variable $x$ in the function with the given number.
Compute the value of $b^x$ . This means raising the base $b$ to the power of $x$ .
If there is a coefficient $a$ in front of the base, multiply the result of $b^x$ by $a$ . If $a$ is $1$ , this step does not change the value.
Simplify the expression if necessary. This could involve performing any additional multiplication or addition/subtraction if the function has more terms.

Let $f(x)=5(3)^x+1$ . Evaluate $f(2)$ without using a calculator.

Follow the order of operations. Be sure to pay attention to the parentheses.

\begin{array}{rcll} f (x) & = & 5 (3)^{x + 1} \\ f (2) & = & 5 (3)^{2 + 1} & Substitute x = 2. \\ = & 5 (3)^{3} & Add the exponents. \\ = & 5 (27) & Simplify the power. \\ = & 135 & Multiply. \end{array}

$\begin{array}{rcll} f(x) & = & 5(3)^{x+1} & \\ f(2) & = & 5(3)^{2+1} & \quad \text{Substitute } x = 2. \\ & = & 5(3)^3 & \quad \text{Add the exponents.} \\ & = & 5(27) & \quad \text{Simplify the power.} \\ & = & 135 & \quad \text{Multiply.} \end{array}$

Suppose a particular population of bacteria is known to double in size every $4$ hours. If a culture starts with $1000$ bacteria, the number of bacteria after $4$ hours is $n(4)=1000·2$ . The number of bacteria after $8$ hours is $n(8)=n(4)·2=1000·2^2$ .

In general, the number of bacteria after $4m$ hours is $n(4m)=1000·2^m$ . Letting $t=4m$ , we see that the number of bacteria after $t$ hours is $n(t)=1000·2^{t/4}$ .

Find the number of bacteria after $6$ hours, $10$ hours, and $24$ hours.

The number of bacteria after $6$ hours is given by $n(6)=1000·2^{6/4} \approx 2828$ bacteria.

The number of bacteria after $10$ hours is given by $n(10)=1000·2^{10/4} \approx 5657$ bacteria.

The number of bacteria after $24$ hours is given by $n(24)=1000·2^{24/4}=1000·2^6=64,000$ bacteria.

Laws of Exponents

The Laws of Exponents are fundamental rules that govern the operations involving powers. These rules are essential for simplifying expressions and are foundational for higher-level math.

laws of exponents

The Product of Powers rule states that when you multiply two exponents with the same base, you can add the exponents.
$b^x·b^y=b^{x+y}$
The Quotient of Powers rule tells us that when dividing exponents with the same base, we subtract the exponents.
$\large\frac{b^x}{b^y} \normalsize = b^{x-y}$
The Power of a Power rule shows that when taking an exponent to another exponent, we multiply the exponents.
$(b^x)^y=b^{xy}$
The Power of a Product rule lets us know that when raising a product to an exponent, each factor in the product is raised to the exponent.
$(ab)^x=a^x b^x$
The Power of a Quotient rule indicates that when a quotient is raised to an exponent, both the numerator and the denominator are raised to the exponent.
$\dfrac{a^x}{b^x} =\left(\dfrac{a}{b}\right)^x$

Note: This is true for any constants $a>0, \, b>0$ , and for all $x$ and $y$

Use the laws of exponents to simplify each of the following expressions.

$\large \frac{(2x^{2/3})^3}{(4x^{-1/3})^2}$
$\large \frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}$

We can simplify as follows:
$\large \frac{(2x^{2/3})^3}{(4x^{-1/3})^2} \normalsize = \large \frac{2^3(x^{2/3})^3}{4^2(x^{-1/3})^2} \normalsize = \large \frac{8x^2}{16x^{-2/3}} \normalsize = \large \frac{x^2x^{2/3}}{2} \normalsize = \large \frac{x^{8/3}}{2}$
We can simplify as follows:
$\large \frac{(x^3y^{-1})^2}{(xy^2)^{-2}} \normalsize = \large \frac{(x^3)^2(y^{-1})^2}{x^{-2}(y^2)^{-2}} \normalsize = \large \frac{x^6y^{-2}}{x^{-2}y^{-4}} \normalsize = x^6x^2y^{-2}y^4 = x^8y^2$

Watch the following video to see the worked solution to this example.

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When you encounter a negative exponent on a term in the denominator of a fraction, you can transform it into a positive exponent by moving the term to the numerator.

\frac{1}{a^{-} n} = a^{n}

$\frac{1}{a^-n}=a^{n}$

Using this rule can significantly simplify expressions involving exponents.