Question

Solve each equation. $$5^{-|x|}=\frac{1}{25}$$

   Solve each equation.
$$5^{-|x|}=\frac{1}{25}$$

Calculus with Applications

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 2, Problem 23

Instant Answer

Solved by Expert Bobby Barnes

03/17/2020

Step 1

Since 25 is 5 squared, we can rewrite $\frac{1}{25}$ as $5^{-2}$. So, the equation becomes: $$5^{-|x|}=5^{-2}$$ Show more…

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Solve each equation. $$5^{-|x|}=\frac{1}{25}$$

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Key Concepts

Absolute Value Equations

Absolute value equations involve solving for the magnitude of a variable, where the absolute value of a number represents its distance from zero on the number line. When an equation is given in the form |x| = k with k being positive, it has two solutions: x = k and x = -k, reflecting the symmetric nature of the absolute value function.

Exponential Equations

Exponential equations are those where variables appear in the exponent position. The key technique for solving such equations is to express both sides of the equation using the same base so that the exponents can be directly equated.

Properties of Exponents

Properties of exponents, such as the rule that a negative exponent signifies the reciprocal of the base raised to the corresponding positive exponent (a^(?m) = 1/(a^m)), are important for rewriting expressions. This allows one to transform a fraction into an equivalent exponential expression.

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