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Chapter 4: Problem 102

Solve each equation. $$ 3^{x^{2}-12}=9^{2 x} $$

Short Answer

Expert verified
The solutions to the equation are \(x = 6\) and \(x = -2\).

Step by step solution

01

Write both sides of the equation with the same base

Manipulate the equation so that both sides are of the same base. Since 9 is 3 squared, replace 9 with \(3^{2}\) in the equation. So, the equation becomes \(3^{x^{2} - 12} = (3^{2})^{2x}\]
02

Simplify the equation

By the rules of exponents, you can multiply the exponents when they're inside brackets. Therefore, the equation can be rewritten as \(3^{x^{2} -12} = 3^{4x}\)
03

Equate the Exponents

With the same bases on either side of the equation, you can simply equate the exponents to each other, giving us \(x^{2} - 12 = 4x\)
04

Rearrange the resulting equation

The resulting equation \(x^{2} - 12 = 4x\) is a quadratic equation. Rearrange the terms to bring them all to one side to get \(x^{2} - 4x - 12 = 0\)
05

Solve the quadratic equation

Solve the quadratic equation \(x^{2} - 4x - 12 = 0\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), or factoring if possible. When factoring the quadratic equation, it becomes \((x - 6)(x + 2) = 0\), and we find that \(x = 6\) or \(x = -2\) - both are the solutions to the equation.

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Most popular questions from this chapter

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ \log _{3}(4 x-7)=2 $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } \log (x+3)=2, \text { then } e^{2}=x+3 $$

Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.
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