Grouping exponential functions according to their bases

Algebra offers three classifications for the base of an exponential function, due to the fact that the numbers used as bases appear to react in distinctive ways when raised to positive powers:

• When , the values of grow larger as x gets bigger — for instance, , , and so on.

• When , the values of show no movement. Raising the number 1 to higher powers always results in the number , , , and so on. You see no exponential growth or decay.

  In fact, some mathematicians leave the number 1 out of the listing of possible bases for exponentials. Others leave it in as a bridge between functions that increase in value and those that decrease in value. It’s just a matter of personal taste.

• When , the value of grows smaller as x gets bigger. The base b has to be positive, and the numbers are all proper fractions (fractions with the numerators smaller than the denominators). Look at what happens to a fractional base when you raise it to the second, fifth, and eighth degrees: . The numbers get smaller and smaller as the powers get bigger.

Adding a coefficient of k to the exponent (making it kx) can change the behavior of the function, depending on what sign k is. After you take into account the effect of multiplying by k, though, the basic properties remain the same.

Grouping exponential functions according to their exponents

An exponent placed with a number can affect the expression that contains the number in somewhat predictable ways. The exponent makes the result take on different qualities, depending on whether the exponent is greater than zero, equal to zero, or smaller than zero:

• When the base and the exponent , the values of get bigger and bigger as x gets larger — for instance, and . You say that the values grow exponentially.
• When the base and the exponent , the only value of you get is 1. The rule is that for any number except . So, an exponent of zero really flattens things out.
• When the base and the exponent — a negative number — the values of get smaller and smaller as the exponents get further and further from zero. Take these expressions, for example: and . These numbers can get very small very quickly.

When bases are related

Many times, bases are related to one another by being powers of the same number.

For example, to solve the equation , you need to write both the bases as powers of 2 and then apply the rules of exponents (see Chapter 4). Here are the steps of the solution:

1. Change the 4 and 8 to powers of 2.
   
2. Raise a power to a power.
   
3. Equate the two exponents, because the bases are now the same, and then solve for x.
   
4. Check your answer in the original equation.
   

When other operations are involved

Changing all the bases in an equation to a single base is especially helpful when you have operations such as roots, multiplication, or division involved.

If you want to solve the equation for x, for example, your best approach is to change each of the bases to a power of 3 and then apply the rules of exponents (see Chapter 4).

Here’s how you change the bases and powers in the example equation (Steps 1 and 2 from the list in the previous section): You change the bases 9 and 27 to powers of 3, replace the radical with a fractional exponent, and then raise the powers to other powers by multiplying the exponents.

When you divide two numbers with the same base, you subtract the exponents. After you have a single power of 3 on each side, you can equate the exponents and solve for x:

Taking out a greatest common factor

When you solve a quadratic equation by factoring out a greatest common factor (GCF), you write that greatest common factor outside the parentheses and show all the results of dividing by it inside the parentheses.

In the equation , for example, you factor from each term and get . After factoring, you use the multiplication property of zero by setting each of the separate factors equal to zero. (If the product of two numbers is zero, at least one of the numbers must be zero; see Chapter 1.) You set the factors equal to zero to find out what x satisfies the equation:

has no solution; 3 raised to a power can’t be equal to 0.

Setting , you change the 9 to a power of 3 to get . Move the to the right, and the equation becomes . Setting the exponents equal to one another, . You find only one solution to the entire equation.

Factoring like a quadratic trinomial

A quadratic trinomial has a term with the variable squared, a term with the variable raised to the first power, and a constant term. This is the pattern you’re looking for if you want to solve an exponential equation by treating it like a quadratic. The trinomial , for example, resembles a quadratic trinomial that you can factor. One option is to consider the quadratic , which would look something like the exponential equation if you replace each with a y. The quadratic in y’s factors into . Using the same pattern on the exponential version, you get the factorization . Setting each factor equal to zero, when , . This equation holds true when , making that one of the solutions. Now, when , you say that , or . In other words, . You find two solutions to this equation: and .

Planning on the future: Target sums

You can figure out what amount of money you’ll have in the future if you make a certain deposit now by applying the compound interest formula. But how about going in the opposite direction? Can you figure out how much you need to deposit in an account in order to have a target sum in a certain number of years? You sure can.

If you want to have $100,000 available 18 years from now, when your baby will be starting college, how much do you have to deposit in an account that earns 5 percent interest, compounded monthly? To find out, you take the compound interest formula and work backward. This time you’ll be solving for the principal (P):

You solve for P in the equation by dividing each side by the value in the parentheses raised to the power 216. According to the order of operations (see Chapter 1), you raise to a power before you multiply or divide. So, after you set up the division to solve for P, you raise what’s in the parentheses to the 216th power and divide the result into 100,000:

A deposit of almost $41,000 will result in enough money to pay for college in 18 years (not taking into account the escalation of tuition fees). You may want to start talking to your baby about scholarships now !

If you have a target amount of money in mind and want to know how many years it will take to get to that level, you can use the formula for compound interest and work backward. Put in all the specifics — principal, rate, compounding time, the amount you want — and solve for t. You may need a scientific calculator and some logarithms to finish it out, but you’ll do whatever it takes to plan ahead, right?

Measuring the actual compound: Effective rates

When you go into a bank or credit union, you see all sorts of interest rates posted. You may have noticed the words “nominal rate” and “effective rate” in previous visits. The nominal rate is the named rate, or the value entered into the compounding formula. The named rate may be 4 percent or 7.5 percent, but that value isn’t indicative of what’s really happening because of the compounding. The effective rate represents what’s really happening to your money after it compounds. A nominal rate of 4 percent translates into an effective rate of 4.074 percent when compounded monthly. This may not seem like much of a difference — the effective rate is about 0.07 higher — but it makes a big difference if you’re talking about fairly large sums of money or long time periods.

You compute the effective rate by using the middle portion of the compound interest formula: . To determine the effective rate of 4 percent compounded monthly, for example, you use .

The 1 before the decimal point in the answer indicates the original amount. You subtract that 1, and the rest of the decimals are the percentage values used for the effective rate.

Table 10-1 shows what happens to a nominal rate of 4 percent when you compound it different numbers of times per year.

Expanding expressions with log notation

You write logarithmic expressions and create logarithmic functions by combining all the usual algebraic operations of addition, subtraction, multiplication, division, powers, and roots. Expressions with two or more of these operations can get pretty complicated. A big advantage of logs, though, is their properties. Because of the special features of log properties, you can change multiplication to addition and powers to products. Put all the log properties together, and you can change a single complicated expression into several simpler terms.

If you want to simplify by using the properties of logarithms, for example, you first use the property for the log of a quotient and then use the property for the log of a product on the first term you get (see Table 10-2 to review these properties):

The last step is to use the log of a power on each term, changing the radical to a fractional exponent first:

The three new terms you create are each much simpler than the original expression.

Rewriting for compactness

Results of computations in science and mathematics can involve sums and differences of logarithms. When this happens, experts usually prefer to have the answers written all in one term, which is where the properties of logarithms come in. You apply the properties in just the opposite way that you break down expressions for greater simplicity (see the previous section). Instead of spreading the work out, you want to create one compact, complicated expression.

To simplify , for example, you first apply the property involving the natural log (ln) of a power to all three terms (see the section “Meeting the properties of logarithms”). You then factor out from the last two terms and write them in a bracket:

You now use the property involving the ln of a product on the terms in the bracket, change the exponent to a radical, and use the property for the ln of a quotient to write everything as the ln of one big fraction:

The expression is messy and complicated, but it sure is compact.

Identifying a rise or fall

You can tell whether a graph will feature exponential growth or decay by looking at the equation of the function.

In order to determine whether a function represents exponential growth or decay, you look at its base:

• If the exponential function has a base , the graph of the function rises as you read from left to right, meaning you observe exponential growth.
• If the exponential function has a base , the graph falls as you read from left to right, meaning you observe exponential decay.

The values of the functions and , for example, both rise as you read from left to right because their bases are greater than 1. The graphs of and both fall as you look at increasing values of x, because 0.2 and 0.9 are both between 0 and 1.

Sketching exponential graphs

In general, exponential functions have no x-intercepts, but they do have single y-intercepts. The exception to this rule is when you change the function equation by subtracting a number from the exponential term; this action drops the curve down below the x-axis.

To find the y-intercept of an exponential function, you set and solve for y. If you want to find the y-intercept of , for example, you replace x with 0 to get . So, the y-intercept is (0, 3). This function rises from left to right because the base is greater than one. The multiplier 0.4 on the x in the exponent acts like the slope of a line — in this case, making the graph rise more slowly or gently (see Chapter 2).

Before you try to graph an equation, you should find another point or two for help with the shape. For instance, if in the previous example, . So, the point (5, 12) falls on the curve. Also, if , (see Chapter 4 for info on dealing with negative exponents). The point is also on the curve. Figure 10-2 shows the graph of this example curve with the intercept and points drawn in.

To graph the function , you first find the y-intercept. When , . So, the y-intercept is (0, 10). Two other points you may decide to use are (1, 9) and . The graph of this function falls as you read from left to right because the base is smaller than one. Figure 10-3 shows the graph of this example curve and the random points of reference.

Graphing log functions with the use of intercepts

When you graph a log function, you look at its x-intercept and the base of the function:

• If the base is a number greater than one, the graph rises from left to right.
• If the base is between zero and one, the graph falls as you go from left to right.

For instance, the graph of has an x-intercept of (1, 0) and rises from left to right. You get the intercept by letting and solving the equation for x. You should choose a couple other points on the curve to help with shaping the graph. The graph of contains the points (2, 1) and . You compute those points by substituting the chosen x value into the function equation and solving for y. (If you need a refresher on how to solve those equations, refer to the section “Solving Logarithmic Equations.”) Figure 10-5 shows you the graph of the example function (the points are indicated).

Reflecting on inverses of exponential functions

Exponential and logarithmic functions are inverses of one another. You may have noticed in previous sections that the flattened, C-shaped curves of log functions look vaguely familiar. In fact, they’re mirror images of the graphs of exponential functions.

An exponential function, , and its logarithmic inverse, , have graphs that mirror one another over the line .

For example, the exponential function has a y-intercept of (0, 1), the x-axis as its horizontal asymptote, and the plotted points (1, 3) and . You can compare that function to its inverse function, , which has an x-intercept at (1, 0), the y-axis as its vertical asymptote, and the plotted points (3, 1) and . Figure 10-6 shows you both of the graphs and some of the points.

The symmetry of exponential and log functions about the diagonal line is very helpful when graphing the functions. For instance, if you want to graph and don’t want to mess with its fractional base, you can graph instead and flip the graph over the diagonal line to get the graph of the log function. The graph of contains the points (0, 1), , and . These points are easier to compute than the log values. You just reverse the coordinates of those points to (1, 0), , and ; you now have points on the graph of the log function. Figure 10-7 illustrates this process.

Notice that the exponential function and its inverse log function cross the line at the same point. This is true of all functions and their inverses — they cross the line in the same place or places. Keep this in mind when you’re graphing.