Solve each equation. (1/e)-x = (1/e2)x+1

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Problem 91

Chapter 5, Problem 91

Solve each equation. (1/e)^-x= (1/e²)^x+1

Verified step by step guidance

Rewrite the bases on both sides of the equation to have the same base. Note that $\frac{1}{e} = e^{-1}$, so rewrite the left side as $\left(e^{-1}\right)^{-x}$ and the right side as $\left(e^{-2}\right)^{x+1}$.

Apply the power of a power rule, which states that $\left(a^m\right)^n = a^{m \cdot n}$, to simplify both sides: the left side becomes $e^{-1 \cdot (-x)} = e^{x}$, and the right side becomes $e^{-2 \cdot (x+1)} = e^{-2x - 2}$.

Since the bases are the same (both are $e$), set the exponents equal to each other: $x = -2x - 2$.

Solve the resulting linear equation for $x$ by first adding \$2x$ to both sides to get $x + 2x = -2$, which simplifies to $3x = -2$.

Divide both sides by 3 to isolate $x$, giving $x = \frac{-2}{3}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Exponents

Understanding how to manipulate exponents is essential, including rules like (a^m)^n = a^(mn) and a^(-m) = 1/a^m. These properties allow simplification and rewriting of expressions to solve equations involving powers.

Exponential Equations

An exponential equation involves variables in the exponent. Solving such equations often requires rewriting both sides with the same base or applying logarithms to isolate the variable.

Equality of Exponential Expressions

If two exponential expressions with the same positive base are equal, their exponents must be equal. This principle allows setting the exponents equal to each other to solve for the unknown variable.

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