Solve for x in both exact and approximate form: a. 5=(10)/(1+9 e^-0.5 x) b. 345=5 e^0.4 x+75

Name: Problem 40, Chapter 9: Precalculus
Uploaded: 2020-12-18T13:48:37-08:00
Duration: 3 min 47 s
Channel: Harmender Singh Yadav
Description: Solve for x in both exact and approximate form: a. 5=(10)/(1+9 e^-0.5 x) b. 345=5 e^0.4 x+75

step 1 This question consists of two parts. We will start with part A. 5 equals to 10 divided by 1 plus

Solve for x in both exact and approximate form: a....

Key Concepts

Exponential Equations

This concept involves equations where the variable appears in the exponent. In these equations, the unknown is entangled within an exponential function, and solving them typically requires isolating the exponential term followed by applying logarithms to extract the variable from the exponent. Such equations are common in modeling growth and decay processes and are fundamental in understanding exponential behavior in various contexts.

Logarithms

Logarithms are the inverse functions of exponentials and serve as a key tool in solving exponential equations. By taking the logarithm of both sides of an equation, one can transform an exponential expression into a linear form with respect to the variable, enabling its isolation and solution. The properties of logarithms, such as the product, quotient, and power rules, are essential for manipulating and simplifying these transformed equations.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations using operations like addition, subtraction, multiplication, division, and factoring. This skill is essential in isolating the exponential term before applying logarithms in exponential equations. By effectively manipulating the equation, the variable can be isolated, allowing for subsequent exact and approximate solutions.

Exact and Approximate Solutions

Exact solutions are expressed symbolically, often using logarithms or other algebraic expressions, and provide a precise form of the answer. Approximate solutions, on the other hand, give numerical estimates by evaluating the exact forms to a required degree of accuracy. Recognizing when to use each approach is important, especially when dealing with exponential and logarithmic equations where the exact form may not be as immediately interpretable as a numerical answer.

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