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Chapter 4: Problem 48

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$2^{2 x}+2^{x}-12=0$$

Short Answer

Expert verified
The solution to the equation \(2^{2x} + 2^x - 12 = 0\) in terms of natural logarithms is \(x = ln(3) \over ln(2)\). The decimal approximation, correct to two decimal places, for \(x\) is 1.58.

Step by step solution

01

Rewrite the Equation

Rewrite the equation to identify it as an equivalent quadratic equation. Let \(u = 2^x\). So, the equation \(2^{2x} + 2^{x} - 12 = 0\) becomes \(u^2 + u - 12 = 0\).
02

Solve the Quadratic Equation

Solve the quadratic equation \(u^2 + u - 12 = 0\) by factoring or using the quadratic formula \(-b \pm sqrt{b^2 - 4ac} \over 2a\). Factoring the equation gives us \((u-3)(u+4)=0\). So, \(u = 3\) or \(u = -4\).
03

Substitute Back and Solve the Exponential Equations

Substitute back \(2^x\) for \(u\). This gives us \(2^x = 3\) and \(2^x = -4\). Solve these equations for \(x\). The equation \(2^x = -4\) has no solution because \(2^x\) is always greater than zero. As for the equation \(2^x = 3\), taking the natural logarithm of both sides gives us \(x = ln(3) \over ln(2)\).
04

Calculate Decimal Approximation

Finally, calculate the decimal approximation for the solution using a calculator. Make sure to round the answer off to two decimal places.

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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. $$3^{x}=2 x+3$$

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

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