Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 3^1-x=1/27

Verified step by step guidance

1. Recognize that the equation is $3^{1\; -\; x}=\frac{1}{27}$.
2. Express the right side, 1/27, as a power of 3. Since 27 = 3^3, then 1/27 = 3^{-3}.
3. Rewrite the equation with the same base: $3^{1\; -\; x}=3^{-3}$.
4. Since the bases are the same and the equation holds, set the exponents equal: $1\; -\; x\; =\; -3$.
5. Solve the resulting linear equation for $x$ by isolating the variable.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position. Solving these equations often requires rewriting expressions so that both sides have the same base, allowing the exponents to be set equal to each other.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is essential to rewrite each side as a power of the same base. For example, 1/27 can be expressed as 3^(-3) since 27 is 3^3, and the reciprocal corresponds to a negative exponent.

Equating Exponents

Once both sides of an equation have the same base, the exponents can be set equal to each other because exponential functions with the same base are one-to-one. This step simplifies the problem to solving a linear equation in the exponent.