Introduction to Differential Equations: Background You’ll Need 3

  • Apply the properties of exponents to simplify exponential expressions

Exponential Functions

Any function of the form [latex]f(x)=b^x[/latex], where [latex]b>0, \, b \ne 1[/latex], is an exponential function with base [latex]b[/latex] and exponent [latex]x[/latex]. Exponential functions have constant bases and variable exponents.

exponential function

For any real number [latex]x[/latex], an exponential function is a function with the form

[latex]f(x)=ab^x[/latex]

where,

  • [latex]a[/latex] is a non-zero real number called the initial value and
  • [latex]b[/latex] is any positive real number ([latex]b>0[/latex]) such that [latex]b≠1[/latex].

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

Let [latex]f(x)=2^x[/latex]. What is [latex]f(3)[/latex]?

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

 

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

Let [latex]f(x)=30(2)^x[/latex]. What is [latex]f(3)[/latex]?

[latex]\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}[/latex]

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

[latex]f(3)=30(2)^3≠60^3=216,000[/latex]
How To: Evaluating Exponential Functions

  1. Given an exponential function, identify [latex]a[/latex], [latex]b[/latex], and the value of [latex]x[/latex] you’re being asked to substitute into the function.
  2. Replace the variable [latex]x[/latex] in the function with the given number.
  3. Compute the value of [latex]b^x[/latex]. This means raising the base [latex]b[/latex] to the power of [latex]x[/latex].
  4. If there is a coefficient [latex]a[/latex] in front of the base, multiply the result of [latex]b^x[/latex] by [latex]a[/latex]. If [latex]a[/latex] is [latex]1[/latex], this step does not change the value.
  5. Simplify the expression if necessary. This could involve performing any additional multiplication or addition/subtraction if the function has more terms.
Let [latex]f(x)=5(3)^x+1[/latex]. Evaluate [latex]f(2)[/latex] without using a calculator.

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

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

Find the number of bacteria after [latex]6[/latex] hours, [latex]10[/latex] hours, and [latex]24[/latex] hours.

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

  1. The Product of Powers rule states that when you multiply two exponents with the same base, you can add the exponents.
    [latex]b^x·b^y=b^{x+y}[/latex]
  2. The Quotient of Powers rule tells us that when dividing exponents with the same base, we subtract the exponents.
    [latex]\large\frac{b^x}{b^y} \normalsize = b^{x-y}[/latex]
  3. The Power of a Power rule shows that when taking an exponent to another exponent, we multiply the exponents.
    [latex](b^x)^y=b^{xy}[/latex]
  4. 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.
    [latex](ab)^x=a^x b^x[/latex]
  5. 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.
    [latex]\dfrac{a^x}{b^x} =\left(\dfrac{a}{b}\right)^x[/latex]

Note: This is true for any constants [latex]a>0, \, b>0[/latex], and for all [latex]x[/latex] and [latex]y[/latex]

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

  1. [latex]\large \frac{(2x^{2/3})^3}{(4x^{-1/3})^2}[/latex]
  2. [latex]\large \frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}[/latex]

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.

[latex]\frac{1}{a^-n}=a^{n}[/latex]

Using this rule can significantly simplify expressions involving exponents.