Solve 2^{x+1} = 3^{2x-1}. Give x in closed form.

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Multiple Choice

Solve 2^{x+1} = 3^{2x-1}. Give x in closed form.

Explanation:
When an equation has the variable in the exponent, use logarithms to bring the exponents down as multiples. Take natural logs of both sides: (x+1) ln 2 = (2x-1) ln 3. Expand and collect x terms: x ln 2 + ln 2 = 2x ln 3 - ln 3 x(ln 2 - 2 ln 3) = - (ln 2 + ln 3) = - ln 6. Now simplify the denominator: ln 2 - 2 ln 3 = ln(2) - ln(9) = ln(2/9) = - ln(9/2). Therefore, x = (- ln 6) / (ln 2 - 2 ln 3) = ln 6 / ln(9/2). So the closed-form answer is x = ln(6) / ln(9/2), which is also x = log base (9/2) of 6.

When an equation has the variable in the exponent, use logarithms to bring the exponents down as multiples. Take natural logs of both sides:

(x+1) ln 2 = (2x-1) ln 3.

Expand and collect x terms:

x ln 2 + ln 2 = 2x ln 3 - ln 3

x(ln 2 - 2 ln 3) = - (ln 2 + ln 3) = - ln 6.

Now simplify the denominator: ln 2 - 2 ln 3 = ln(2) - ln(9) = ln(2/9) = - ln(9/2). Therefore,

x = (- ln 6) / (ln 2 - 2 ln 3) = ln 6 / ln(9/2).

So the closed-form answer is x = ln(6) / ln(9/2), which is also x = log base (9/2) of 6.

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