Question

Solve each equation. $$ 0.3\left(4^{0.2 x}\right)=0.2 $$

   Solve each equation.
$$
0.3\left(4^{0.2 x}\right)=0.2
$$

Calculus, Early Transcendentals

Calculus, Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 0, Problem 54

Instant Answer

Solved by Expert Gregory Higby

07/26/2021

Step 1

3 to isolate the term with the variable x on one side of the equation: $$ 4^{0.2x} = \frac{0.2}{0.3} = \frac{2}{3} $$ Show more…

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Solve each equation. $$ 0.3\left(4^{0.2 x}\right)=0.2 $$

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Key Concepts

Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. Solving these equations typically involves isolating the exponential expression and then applying logarithms to both sides in order to 'bring down' the exponent so that the variable can be solved using algebraic methods.

Logarithms

Logarithms are the inverse functions of exponential functions and are essential tools for solving equations that contain exponentials. By taking the logarithm of both sides of an exponential equation, one can use properties such as the power rule to simplify and solve for the variable in the exponent.

Properties of Exponents

The properties of exponents, such as the ability to rewrite expressions by manipulating powers and bases, are fundamental when handling exponential equations. These properties enable the simplification of expressions and the isolation of terms, paving the way for the use of logarithms in solving the equation.

Algebraic Manipulation

Algebraic manipulation, including operations like division and the rearrangement of terms, is crucial for isolating the exponential part of the equation. Mastery of these techniques is necessary to set up the equation in a form that allows the application of logarithms or other methods to solve for the unknown variable.

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