Exponential and Logarithmic Functions: Learn It 2

Exponential Functions Cont.

Evaluating Exponential Functions

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

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

f(x)=2xf(3)=23Substitute x=3.=8Evaluate the power.

 

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)?

f(x)=30(2)xf(3)=30(2)3Substitute x=3.=30(8)Simplify the power first.=240Multiply.

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

f(3)=30(2)3603=216,000

How To: Evaluating Exponential Functions

  1. Given an exponential function, identify a, b, and the value of x you’re being asked to substitute into the function.
  2. Replace the variable x in the function with the given number.
  3. Compute the value of bx. This means raising the base b to the power of x.
  4. If there is a coefficient a in front of the base, multiply the result of bx by a. If a is 1, 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 f(x)=5(3)x+1. Evaluate f(2) without using a calculator.


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=1000·22.

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

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

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

  1. The Product of Powers rule states that when you multiply two exponents with the same base, you can add the exponents.
    bx·by=bx+y
  2. The Quotient of Powers rule tells us that when dividing exponents with the same base, we subtract the exponents.
    bxby=bxy
  3. The Power of a Power rule shows that when taking an exponent to another exponent, we multiply the exponents.
    (bx)y=bxy
  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.
    (ab)x=axbx
  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.
    axbx=(ab)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.

  1. (2x2/3)3(4x1/3)2
  2. (x3y1)2(xy2)2

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.

1an=an

Using this rule can significantly simplify expressions involving exponents.