Solving Exponential Equations Algebraically
Write the exponential expressions with the same base.The bases are 2 and 4. Since 4 can be written as , substitute for 4.
Use the equality property.The expressions have the same base, so the exponents are equal.
Use technology to evaluate the logarithm, and simplify the expression for .If needed, use the change of base rule to rewrite the logarithm with a base that a calculator can compute, such as base 10 or base .
Isolate the exponential expression.Divide both sides of the equation by 5.
Write the equation in logarithmic form.The base of the exponential expression is , so use the natural logarithm.
Exponential equations are used in a wide variety of scientific fields to model data and make predictions about the natural world. For example, radiocarbon dating is based on the application of an exponential model.
Radiocarbon dating is a method of approximating the age of an object that contains organic material. Carbon-14 is a radioactive isotope of carbon that decays into another element over time. By measuring the amount of carbon-14 left in an artifact and comparing that to the amount that would be expected in a living organism, scientists can determine how much carbon-14 has decayed and then calculate how much time it would have taken for that amount of carbon-14 to decay. This gives an estimate of how old the object is.
Take the logarithm of both sides of the equation. Then simplify.The base of the exponential expression is , so use the natural logarithm.
Solving Exponential Equations Using Substitution
Use a temporary variable to represent an exponential expression.Substitute for .