Solving Exponential And Logarithmic Equations

Solving Exponential Equations . An exponential equation 15 is an equation that includes a variable as one of its exponents. In this section we describe two methods for solving exponential equations. First, recall that exponential functions defined by 92f x bx92 where 92b gt 092 and 92b 192, are one-to-one each value in the range corresponds to exactly one element in the domain.

To solve exponential equations, rewrite each side to have the same base, allowing you to set the exponents equal.For example, if 2 x 16, rewrite 16 as 2 4 to find x 4.If bases differ, use logarithms to isolate the variable. For logarithmic equations, set logs of the same base equal or convert to exponential form to solve.

We can solve exponential equations with base e, by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions.

Learn how to solve simple and complex exponential and logarithmic equations, and how to use them to model real-life problems of growth and decay. See examples, definitions, properties, and tips for solving different types of equations.

Understanding Exponential Equations. Exponential equations involve expressions where the variable is an exponent. These equations can appear in scientific contexts, such as calculating population growth. Key here are the properties of exponents, which simplify these expressions. Properties of Exponents am 92cdot an amn amn am

Learn how to use index laws and laws of logarithms to simplify and manipulate exponential and logarithmic functions. Find examples, definitions, explanations and interactive activities to help you master these topics.

Section 1.9 Exponential and Logarithm Equations. In this section we'll take a look at solving equations with exponential functions or logarithms in them. We'll start with equations that involve exponential functions. The main property that we'll need for these equations is, 9292log _bbx x92

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations.

Use property 9 to rewrite the above logarithmic equation as follows 2 x e 3 Solve for x x e 3 2 Example 4 Find all real solutions to the equation e 3x - 1. Solution to Example 4 The range of basic exponential functions is 0 , , hence e 3x cannot be negative and therefore the given equation has no real solutions. Example 5

Learn how to convert between exponential and logarithmic forms, and use properties and definitions to solve equations. See examples, practice problems and solutions.