Evaluating an Exponential Model
An exponential model can be specified by an initial value and a base. Evaluating it means substituting a stated input and calculating the corresponding output. The calculation is straightforward when the exponent is an integer, but negative and rational exponents require careful use of exponent laws.
Tables can also provide enough information to recover a missing input or a model parameter. The key is to translate each table entry into an equation, then solve using powers that can be recognized exactly. Here we focus on exact calculations; finding an exponent that is not a recognizable exact power requires logarithms, which are not needed for the methods here.
The positive-base condition allows powers with rational exponents to be interpreted as real values. In exact hand calculations, those powers are most direct when the required roots are exact. The condition \(b\ne 1\) distinguishes an exponential model from a constant function; the evaluation steps themselves still make sense for \(b=1\).
Substitute First, Then Simplify
A reliable calculation starts by replacing every occurrence of \(x\) with the specified input, using parentheses when the input is negative or a fraction. Only then should the exponent be simplified. In particular, the exponent applies to the base, not automatically to the coefficient \(a\).
Worked Example: Evaluating at a Negative Input
Let \(f(x)=5\cdot 2^x\). Find \(f(-2)\), then verify the result by substitution.
Substitute \(-2\) for \(x\):
A negative exponent means take the reciprocal of the corresponding positive power. Thus
Therefore,
Check by substituting the input into the original expression: \(5\cdot 2^{-2}=5\cdot\frac14=\frac54\), as calculated.
This example illustrates an important distinction: the coefficient remains \(5\), while the negative exponent makes \(2^x\) a reciprocal. It does not make the whole expression negative.
Zero and Rational Exponents
The exponent laws give two useful rules for exact evaluation. For any nonzero base \(b\), \(b^0=1\); for a positive base \(b\) and positive integer \(n\), \(b^{-n}=1/b^n\). A rational exponent describes a root and a power. For example, \(b^{m/n}\) is the \(n\)-th root of \(b^m\) when that real root is defined. For positive \(b\), it can be evaluated as \((\sqrt[n]{b})^m\).
Worked Example: Evaluating a Rational Exponent
Let \(g(x)=4\left(\frac{1}{16}\right)^x\). Find \(g\left(-\frac32\right)\).
Substitute the input:
First use the negative exponent to take a reciprocal:
The exponent \(3/2\) means take a square root and then cube, or cube and then take a square root. Using the square root first gives
Now multiply by the coefficient:
Check by simplifying the original substituted expression: \(4(1/16)^{-3/2}=4\cdot 16^{3/2}=4\cdot64=256\). The calculation is consistent.
For a non-integer input, the exact calculation may be less immediate. For instance, if the necessary root is not an integer or a familiar exact value, a decimal approximation may be appropriate. The important point is to preserve the exact expression until the required method of approximation is available, rather than pretending that an unevaluated root is an integer.
Finding an Unknown Input by Matching Powers
To find an unknown input, set the model equal to the stated output. If the coefficient is nonzero, divide both sides by it to isolate the exponential power. Then express the resulting value as a power of the same base, if possible, and compare exponents. This method works when the target value can be written exactly as a power of the base. It does not provide a general method for arbitrary target values.
Worked Example: Finding an Input from an Exact Output
Suppose \(f(x)=3\cdot 2^x\). Find the input \(x\) for which \(f(x)=24\), and check the answer.
Set the model equal to the desired output:
Divide both sides by \(3\):
Since \(8=2^3\), the equation becomes \(2^x=2^3\). Equal powers with the same positive base different from \(1\) have equal exponents, so \(x=3\).
Check by substituting \(x=3\) in the original model:
The output is the required \(24\), so \(x=3\) is correct.
Matching powers is not the same as guessing from a decimal pattern. The target should be rewritten as an exact power, and the resulting candidate should be checked in the original equation. If the target cannot be expressed in a usable exact form, this technique alone may not determine the input.
Recovering Parameters from a Table
A table gives pairs of inputs and outputs. Each row becomes an equation of the form \(f(x)=y\). When the model has two unknown parameters, two suitably chosen rows may determine them. A particularly useful feature is the ratio of outputs at two inputs: the common coefficient cancels, leaving a relation involving only the base.
After determining the base from the ratio, substitute either table entry into the model to find the initial value. The ratio is useful only when the table values and input difference allow the base to be recovered; it does not guarantee that every table determines a unique model.
Worked Example: Finding the Base and Initial Value from a Table
Assume the table comes from \(f(x)=a b^x\), with \(b>0\) and \(b\ne1\).
| \(x\) | \(f(x)\) |
|---|---|
| 1 | 10 |
| 3 | 250 |
Use the ratio of the output at \(x=3\) to the output at \(x=1\):
Substitute the table values:
Since the model requires \(b>0\), \(b=5\). Use \(f(1)=10\) to find \(a\):
The model is \(f(x)=2\cdot5^x\). Check both table entries:
Both calculated outputs match the table, so the recovered parameters are correct.
Finding a Missing Table Entry
The same substitution method applies when a table gives the model parameters or enough information to determine them, and one input or output is missing. Always distinguish between a missing output and a missing input: for a missing output, substitute the input and evaluate; for a missing input, form an equation and solve by matching powers when possible.
Worked Example: Completing a Table by Evaluating
Suppose \(h(x)=2\cdot 3^x\), and a table contains \(h(0)\), \(h(1)\), and \(h(2)\). Find each output and check them directly.
At \(x=0\):
At \(x=1\):
At \(x=2\):
Check the entries by substituting each input into the original formula: the values are \(2\cdot1=2\), \(2\cdot3=6\), and \(2\cdot9=18\), respectively. Thus the completed table is consistent with the model.
Why the Ratio Method Works
The ratio method is effective because both outputs contain the same initial-value factor. Dividing cancels that factor, so the comparison isolates the effect of changing the input. Here is the full algebra for two inputs \(u\) and \(v\), under the conditions \(a\ne0\) and \(f(u)\ne0\):
The last equality uses the quotient law for exponents. This argument also explains why a zero output cannot be used as the denominator in this calculation. In the usual model with nonzero \(a\) and positive \(b\), all outputs are nonzero, but a table should still be checked against the model assumptions rather than used mechanically.
Common Pitfall: Misplacing the Exponent
A frequent error is to treat \(a b^x\) as though the exponent applies to the entire product. In general, \(a b^x\) means multiply \(a\) by \(b^x\); it is not \((ab)^x\). The distinction changes the result.
Worked Example: Correcting an Exponent Error
Evaluate \(p(2)\) for \(p(x)=3\cdot 2^x\). A mistaken calculation might write
This is incorrect because the model has the coefficient \(3\) outside the power. The correct substitution is
Check using the original formula at \(x=2\): it gives \(3\) multiplied by \(2^2=4\), which is \(12\). The incorrect value \(36\) would come from the different expression \((3\cdot2)^2\), not from the stated model.
Using the Method Carefully
When working with a table or a specified input, keep the original model visible until the final check. A short sequence of steps helps prevent errors:
Identify its coefficient, base, and exponent before substituting or solving.
For an output, substitute the input. For a missing parameter or input, write the equation represented by the table entry.
Simplify exact powers, or cancel the common coefficient when comparing two nonzero outputs.
Substitute the proposed answer and verify that it produces the stated value.
These calculations are useful in any setting where an exponential rule is given by a formula or a small set of exact data. They separate the arithmetic of evaluating a model from the later study of its graph or the broader behavior of exponential growth and decay. If exact powers do not reveal an unknown input, do not force a guessed answer; the method has reached its limit, and a different tool is needed.
Check Your Understanding
Work each question by writing the substitution or equation first, and check any proposed answer in the original model.
- If \(q(x)=7\cdot 2^x\), find \(q(-1)\).
- If \(r(x)=4\cdot 3^x\), which input gives the output \(108\)?
- A model \(s(x)=a b^x\) has \(s(0)=5\) and \(s(2)=45\), with \(b>0\). Find \(a\) and \(b\), then check both table values.
- For \(t(x)=2\left(\frac{1}{9}\right)^x\), evaluate \(t\left(-\frac12\right)\).
- Explain why \(5\cdot 2^3\) is not the same expression as \((5\cdot2)^3\).
