Question

Solve the equations. $$\frac{1}{2^{x}}=\frac{5}{8^{x+3}}$$

   Solve the equations.
$$\frac{1}{2^{x}}=\frac{5}{8^{x+3}}$$

Calculus: A Complete Course

Calculus: A Complete Course

Robert Adams ,… 9th Edition

Chapter 3, Problem 13

Instant Answer

Solved by Expert Uma Kumari

10/08/2021

Step 1

This gives us: $$\log \left(\frac{1}{2^{x}}\right) = \log \left(\frac{5}{8^{x+3}}\right)$$ Show more…

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Solve the equations. $$\frac{1}{2^{x}}=\frac{5}{8^{x+3}}$$

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Key Concepts

Exponential Equations

Exponential equations are those in which variables appear as exponents. Solving these requires applying logarithmic or algebraic properties to isolate the exponent variable on one side of the equation. The process often involves rewriting terms so that both sides of the equation have the same base, thereby allowing direct comparison of the exponents.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate expressions involving powers. These include rules like a^(m+n) = a^m * a^n and (a^m)^n = a^(mn), as well as handling negative exponents and rewriting reciprocal values. Understanding these properties is crucial for simplifying expressions and solving equations that involve exponential terms.

Base Conversion

Base conversion involves rewriting different exponential expressions to have the same base. Doing so simplifies the process of solving exponential equations because it enables the direct equating of the exponents. This method is especially useful when dealing with expressions like powers of 2 and 8, since 8 can be written as 2^3.

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