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Chapter 4: Problem 101

Solve each equation. $$5^{x^{2}-12}=25^{2 x}$$

Short Answer

Expert verified
The solutions are \( x = 6\) and \(x = -2\)

Step by step solution

01

Express both sides with the same base

Since both 5 and 25 are powers of 5, express 25 as \(5^{2}\). Then, the equation becomes: \[5^{x^{2}-12}= (5^{2})^{2x}\] or \[5^{x^{2}-12} = 5^{4x} \]
02

Set the exponents equal to each other

Now that both sides have the same base, you can set the exponents equal to each other: \[ x^{2}-12 = 4x \]
03

Rearrange the equation to Quadratic Form

Rearrange the equation into standard quadratic form: \[x^2 - 4x - 12 = 0\]
04

Solve the quadratic equation

Next, solve the quadratic equation, by either factoring, using quadratic formula, or completing the square. Here, we can use factoring to get \((x - 6) (x + 2) = 0\) which yields \(x = 6\) and \(x = -2\)

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Most popular questions from this chapter

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. $$5^{x}=3 x+4$$

Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$

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. $$\log (x-15)+\log x=2$$
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