Question

Solve the exponential equation. Round to three decimal places, when needed. $$3^{-x}=10^{-4 x+1}$$

   Solve the exponential equation. Round to three decimal places, when needed.
$$3^{-x}=10^{-4 x+1}$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 4, Problem 18

Instant Answer

Solved by Expert Kader Bayar

05/07/2020

Step 1

Using the property of logarithms, we can bring the exponents to the front of the logarithm. This gives us: $$-x \log 3 = (-4x+1) \log 10$$ Show more…

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Solve the exponential equation. Round to three decimal places, when needed. $$3^{-x}=10^{-4 x+1}$$

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Key Concepts

Logarithmic Properties

Key properties of logarithms, such as ln(a^b) = b ln(a) and the logarithm of a product being the sum of logarithms, enable the simplification of equations. These properties are instrumental in isolating the variable when it appears in the exponent.

Logarithms

Logarithms are the inverses of exponential functions and are used to solve equations where the variable is in the exponent. By taking the logarithm of both sides of an equation, one can bring the exponent down, thus converting an exponential equation into a linear one in the variable.

Numerical Approximation and Rounding

After solving an equation analytically, the solution may need to be expressed as a numerical approximation. Rounding, which involves adjusting the number to a specified degree of precision, is an essential step when converting an exact solution into a practical answer, especially when the problem requires a certain number of decimal places.

Exponential Equations

Exponential equations are equations in which the unknown variable appears in the exponent. Solving such equations often requires transforming the expression so that the variable can be isolated. Understanding these equations is crucial because they model growth and decay processes, among other phenomena.

Properties of Exponents

The properties of exponents allow us to manipulate expressions where the variable is in the exponent. For instance, recognizing that a^(-x) is the reciprocal of a^(x) and being able to distribute exponents over products are fundamental skills when working with exponential equations.

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