Select the correct alternative from the given choices. Solve...

Select the correct alternative from the given choices.
Solve for $x: 2\left\{3^{2(1+x)}\right\}-4\left(3^{2+x}\right)+10=0$
(A) $-1, \log _{3}\left(\frac{5}{3}\right)$
(B) $-1, \log _{3} 2$
(C) $-1, \frac{5}{3}$
(D) $-1, \log _{3}\left(\frac{3}{5}\right)$

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Key Concepts

Logarithms

Logarithms, being the inverse functions of exponentiation, provide a systematic way to solve equations where the variable is in the exponent. They help isolate the variable by converting the exponential equation into a linear one in terms of logarithms.

Substitution Method

The substitution method involves replacing a repeated complex exponential expression with a single variable. This often converts the original equation, which may involve exponential functions, into a quadratic or another simpler form that can be solved using standard algebraic methods.

Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. They often require rewriting terms so that all exponential parts share a common base, which makes the equation easier to manipulate and solve.

Properties of Exponents

These properties, such as a^(m+n) = a^m * a^n and (a^m)^n = a^(m*n), allow transformation and simplification of exponential expressions. They are crucial for rewriting terms in a form amenable to substitution or algebraic manipulation.

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