How do I solve exponential equations?
Exponential equations in the Earth sciences

An introduction to exponential equations

Throughout the Earth sciences you will find equations that have a variable in the exponent, referred to as exponential equations. These equations are common because they describe dynamic processes such as exponential growth and decay. Exponential equations are used to model exponential growth in populations over time and to model exponential decay of materials like radiogenic isotopes. The rate of decay of radiogenic isotopes is an important tool for determining the age of geologic materials such as rocks and minerals. The graph to the right shows what this decay looks like through time. The general form of exponential equations is y=ab^x. In which, x and y are variables and a and b are constants. X is commonly the independent variable and y is the dependent variable because y is determined based on what you put in for x.

What are examples of exponential equations?

Exponential equations describe dynamic systems and graphing equations may help you visualize how changing x causes rapid change in y. Exponential equations with a positive exponent and where `b>1`, such as `y=2^x`, show exponential growth, and exponential equations with a positive exponent and where `0<b<1`, such as `y=(1/2)^x` show exponential decay. Equations with a negative exponent, like `y=2^-x`, also represent exponential decay. What are some things you notice when looking at the graphs below?

• Both equations make curves instead of straight lines. This highlights that these equations are not linear equations.
• Notice how the exponential equations behave differently as the value for x increases. Change in exponential growth (represented by the y-value) starts slowly and then increases rapidly, whereas for exponential decay, the change in the y-value is large at first and then much smaller as x increases.
• Exponential growth has a positive slope (as x increases, y increases) and exponential decay has a negative slope (as x increases, y decreases).

In the sciences, exponential growth and exponential decay are continuous processes that occur over time. Scientists frequently use a few different constants to model these systems. The constant e is used as the limit of continuous growth or decay, and it equals approximately 2.718. Notice in the equations below, the constant e replaces the constant b. The constant k is the growth or decay constant. For growth, k is positive, and for decay, k is negative. Scientists are mostly interested in how these systems change over time, so time (t) is added to the exponent term and multiplied by k, the growth or decay constant.

• Exponential growth (used for population or nuclear chain reactions): `N=N_0 e^(kt)`
• Exponential decay (used to determine the age of a material from radioactive decay or the amount of a reactant in a reservoir): `N=N_0 e^(-kt)`

How to work with exponents and logarithms

Exponents and logarithms are closely related and can be seen as inverse of each other. Commonly, they are described as "undoing" each other.

• `y=a^x` is an exponential equation stated as "y equals a to the power of x"
• `log_a(y)=x` is a logarithmic equation stated as "log base-a of y equals x". Specifically, `log_a(y)` asks, what exponent do I need on a so that `a^x=y`

This relationship is used to simplify exponential equations like the exponential growth equation:

• `N=N_0 e^(kt)` Take the natural log (ln) of both sides of the equation 
• `lnN=lnN_0 lne^(kt)`
• Because `lne` undoes the operation of raising `e` to the power of `kt` the equation simplifies to:
• `lnN=lnN_0 * kt` 

How do I solve exponential equations?

Most of the steps for solving exponential equations require algebra and working with logarithms to move the variable out of the exponent. Below are examples of some steps to take when you are solving an equation with variables in the exponent. In this module, we are focusing on exponential equations that include the log of e or the natural log.

Exponential growth

Under the following steps you can find an example in which we determine the time (t) until a population reaches a certain size.

In considering resource use and planning for Island County in Washington, which includes the San Juan Islands, it is important to assess population growth trends. Island County has had a 1.3% growth rate since 2000. In 2000, the county had 79,300 residents. In how many years can the county expect to have 100,000 residents?
Use the equation for population growth. 

`N=N_0 e^(kt)` 

N is the population; N₀ is the starting population; k is the growth rate; and t is time. The letter e is a constant that equals approximately 2.718 and represents the limit of continuous growth. 

Step 1. Determine which variables are known and unknown. Usually there is just one unknown.

Step 2. Rearrange the equation to isolate the term with the variable you want to solve for. (For more general help on rearranging equations - check out this page)

Step 3. Use the appropriate logarithm to isolate the variable of interest. Because t is in the exponent, use the matching base logarithm to isolate t.

Step 4. Insert the known values into the rearranged equation.

Step 5. Solve for the unknown.

What does the population growth Island County look like on a line graph?

On the line graph of years vs. population, you can see a graphical example of exponential growth. The line shows the growth rate over time, the purple and green boxes intersect at the area of interest, and the dot represents when the population of Island County will reach 100,000 people.

Thinking more about the math

If the growth rate increased from 1.3% to 2.0%, what would happen to the steepness (slope) of the line in the graph above?

The slope would increase, and the line would be steeper

The slope would decrease, and the line would be less steep The higher the growth rate, the steeper the line. A less steep slope would indicate a lower growth rate

Exponential decay

Earth scientists regularly need to determine the age of a rock to place the rock unit or geologic event in an appropriate temporal context. Minerals within rocks preserve ratios of radiogenic isotopes that can be used to calculate when the mineral crystallized or dropped below a certain temperature. Each radiogenic isotope has a unique decay constant that can be used to determine the isotope's half-life `(t_(1/2))`. The table to the right shows some common examples used in the Earth sciences. Under the following steps you can find an example in which we determine the age of a rock.

Muscovite within a granite originally had 2400 mg of ⁴⁰K and no ⁴⁰Ar. It now has 800 mg of ⁴⁰K. Assuming that the only chemical process is radioactive decay of potassium ⁴⁰K to argon ⁴⁰Ar, what is the age of the granite? The decay constant, λ, for the K-Ar system is `lambda=5.54xx10^(-10)  1/(yrs)`

To determine age, earth scientists use the exponential decay equation, often referred to as the age equation. 

`N=N_0 e^(-lambdat)`

Where N is the amount of isotope present today, N₀ is initial amount of isotope, t is time, and λ is the decay constant.

You may notice that the mass of ⁴⁰K remaining is 33.3%. On the table of percentage of ⁴⁰K, time elapsed in millions of years (Ma), and the number of half-lives, you can see that 33.3% of ⁴⁰K remaining is in between 1.5 and 2.0 half-lives and 1875 and 2500 Ma. As a first approximation, the age of the granite should be between 1875 Ma and 2500 Ma, and it should be closer to 1875 Ma. Now that you have a sense of a reasonable answer, work through the steps to determine the age more precisely.

Step 1. Determine which variables are known and unknown. Usually there is just one unknown.

Step 2. Rearrange the equation to isolate the term with the variable you want to solve for.

Step 3. Use the appropriate logarithm to isolate the variable of interest. Because t is in the exponent, use the matching base logarithm to isolate t.

Step 4. Insert the known values into the rearranged equation.

Step 5. Solve for the unknown.

Where do you solve exponential equations in Earth science?

• Environmental science - Population growth
• Geochronology - Dating Earth materials
• Hydrology - Groundwater contamination
• Geophysics - Earthquake moment magnitude and radar wave attenuation 
• Atmospheric science - Atmospheric pressure

Next steps

I am ready to PRACTICE!
If you think you have a handle on the steps above, click on this bar to try practice problems with worked answers.

Or, if you want even more practice, see More help below.

More help (resources for students)

• Mathworld has some very good resources explaining:
  • The constant e (more info),
  • Exponential decay (more info),
  • Exponential growth (more info), and
  • Graphical and mathematical representations of the exponential function (more info)
• Visionlearning has clear explanations of exponential equations and explanations of math used in science
• OpenAlgebra Solving Exponential Equations
• Graphing Exponentials
• Khan Academy: Exponential and logarithmic functions

Pages written by Rory McFadden (Science Education Resource Center) and Eric Baer (Highline College).