Exponents and Logarithms

How do you write an exponential growth and decay function?

Quick answer

To write an exponential growth and decay function, use the formula y = a(b)^x, where a is the starting amount, b is the base or factor, and x is the number of time periods. For a growth rate r, the base is b = 1 + r. For a decay rate r, the base is b = 1 - r.

The topic explained

The basic exponential growth and decay function

An exponential growth and decay function models any quantity that multiplies by a constant number each period. The standard form is y = a(b)^x. In this formula, y is the final value, a is the initial quantity before any time has passed, b is the growth or decay factor, and x is the number of elapsed time periods.

You can tell growth from decay by looking at the base b. When b is greater than 1, the output grows over time. When b is between 0 and 1, the output decreases over time. The starting amount a must be positive, and b can never be negative or zero.

Converting percentage rates into factors

Word problems often state change as a percentage rather than a factor. First, convert that percentage into a decimal by dividing by 100. Let this decimal be r. If a population grows by 6 percent each year, r = 0.06.

Next, determine whether the problem describes growth or decay. For growth, add the rate to 1 so that b = 1 + r. A 6 percent increase yields a base of 1 + 0.06 = 1.06. For decay, subtract the rate from 1 so that b = 1 - r. A car depreciating by 15 percent per year gives a base of 1 - 0.15 = 0.85.

Step-by-step example from a word problem

Suppose a coffee shop opens with 200 rewards members, and membership grows by 12 percent each month. Identify the parts: the initial amount a is 200, the monthly rate r is 0.12, so the factor b is 1 + 0.12 = 1.12. The function is M(t) = 200(1.12)^t, where t represents months.

To find the membership after 5 months, substitute 5 for t: M(5) = 200(1.12)^5. Calculate 1.12 to the 5th power first to get approximately 1.7623, then multiply by 200. After rounding to the nearest whole person, the shop has about 352 members.

Handling doubling time and half-life

Some problems give a doubling period or a half-life instead of a percentage rate. In doubling problems, the base is always 2. If a culture of 50 bacteria doubles every 3 hours, write B(t) = 50(2)^(t/3). Dividing t by 3 ensures the quantity only doubles when 3 full hours have passed.

In half-life problems, the base is 0.5. If 80 grams of a radioactive substance has a half-life of 14 days, write N(t) = 80(0.5)^(t/14). Always make sure the units for time in your exponent match the time period stated in the problem.

Free printable worksheet

Exponential Growth and Decay Practice

Identify whether each scenario represents growth or decay, write the exponential model y = a(b)^x, and solve for the requested value.

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Personal AI lessons and practice, from preschool to university

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Exponential Growth and Decay PracticeName:

A town with 4,500 residents grows by 5% each year. Identify the initial amount a, the factor b, and write the model for population P(t) after t years.
A new computer bought for $1,200 loses 20% of its value every year. State the decay factor and write the value function V(t).
A savings account starts with $600 and earns 4% interest compounded annually. How much money is in the account after 3 years?
A medication in the bloodstream starts at 250 mg and decreases by 30% each hour. How many milligrams remain after 4 hours? Round to one decimal place.
A sample of 80 yeast cells doubles every 4 hours. Write the function Y(t) in terms of hours t, then calculate the population after 12 hours.
A 100-gram sample of an isotope has a half-life of 6 days. Write the function R(t) and determine how much remains after 18 days.

Common questions

What is the difference between growth rate and growth factor?

The growth rate is the percentage increase expressed as a decimal, such as 0.08 for 8 percent. The growth factor is what you multiply by each cycle, calculated as 1 plus the rate (1 + 0.08 = 1.08). The base in your formula must always be the factor, not just the rate.

What is the most common mistake students make with decay functions?

The most common mistake is using the rate directly as the base instead of subtracting it from 1. For instance, if an item loses 25% of its value, using 0.25 as the base means 75% of the value vanished in the first step. The correct base is 1 - 0.25 = 0.75.

Why is the exponent divided by a number in half-life problems?

Dividing the time variable by the cycle length scales the clock. If a half-life is 5 days, the exponent t/5 equals 1 only after 5 full days have passed. This ensures the substance halves once every 5 days rather than every single day.

What topics follow exponential functions in 11th grade math?

Students typically study logarithms next. Logarithms are the inverses of exponential functions and give you the algebraic tools to solve for an unknown exponent, such as finding exactly how many years it takes for an investment to triple.

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