Question

Use logarithms to solve. $3 e^{3-3 x}+6=-31$

   Use logarithms to solve.
$3 e^{3-3 x}+6=-31$

Precalculus

Jay Abramson 1st Edition

Chapter 4, Problem 318

Instant Answer

Solved by Expert Sarah Stryke

03/18/2020

Step 1

This gives us: \[3 e^{3-3 x} = -31 - 6 = -37\] Show more…

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Use logarithms to solve. $3 e^{3-3 x}+6=-31$

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Key Concepts

Domain Considerations

When solving equations involving exponentials and logarithms, it is important to consider the domain restrictions of these functions. Exponential functions always yield positive outputs, so if an equation suggests that an exponential term equals a negative number, it indicates that there may be no real solutions. This insight helps in determining the existence of valid solutions.

Properties of Logarithms

Understanding the fundamental properties of logarithms—such as the power rule, product rule, and quotient rule—is essential when simplifying expressions after applying logarithms. The power rule, in particular, lets you bring the exponent in front of the logarithm as a multiplier, thereby transforming the equation into a more manageable algebraic equation.

Exponential Equations

Exponential equations are those in which the unknown variable appears in the exponent. To solve such equations, one typically isolates the exponential expression and then applies logarithms to both sides, making it possible to bring the variable down from the exponent so that it can be solved using algebraic methods.

Logarithms and Their Inverse Relationship

Logarithms serve as the inverse functions of exponentials. This crucial relationship allows us to transform equations from an exponential form to a form where the variable appears as a coefficient. By applying logarithms to both sides of an equation, the exponential operation is reversed, facilitating the solution of the variable.

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