MATHEMATICS MIDDLE SCHOOL

Explain how solve 4^(x+3)=7 using the change of base formula log base b of y equals log y over log b. Include the solution for x in your answer. Round your answer to the nearest thousandth.

Answers

Answer 1

Answer:

The value of x is -1.596

Solution:

Given equation is:

$4^((x+3)) = 7$

Let us solve using change of base formula log base b of y equals log y over log b

From given,

$4^((x+3)) = 7$

$\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)$

Therefore,

$\ln \left(4^(x+3)\right)=\ln \left(7\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$

$\ln \left(4^(x+3)\right)=\left(x+3\right)\ln \left(4\right)\n\n\left(x+3\right)\ln \left(4\right)=\ln \left(7\right)\n$

Let us simplify the above

$\left(x+3\right)\cdot \:2\ln \left(2\right)=\ln \left(7\right)\n\n\mathrm{Divide\:both\:sides\:by\:}2\ln \left(2\right)\n\n(\left(x+3\right)\cdot \:2\ln \left(2\right))/(2\ln \left(2\right))=(\ln \left(7\right))/(2\ln \left(2\right))\n\n$

$\mathrm{Simplify}\n\nx+3=(\ln \left(7\right))/(2\ln \left(2\right))\n\n\mathrm{Subtract\:}3\mathrm{\:from\:both\:sides}\n\nx+3-3=(\ln \left(7\right))/(2\ln \left(2\right))-3\n\n\mathrm{Simplify}\n\nx=(\ln \left(7\right))/(2\ln \left(2\right))-3$

Substitute the values

ln 7 = 1.9459101490553132

ln 2 = 0.6931471805599453

Therefore,

$x = (1.9459101490553132)/(2 * 0.6931471805599453) - 3\n\nx = 1.40367746103 - 3\n\nx = -1.59632253897 \approx -1.596$

Thus solution for x is found

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HELP PLEASE Sam and George both worked hard over the summer. Together they earned a total of $525. George earned$75 more than Sam.
a. Write a system of equations for the situation. Use s for the amount Sam earned and g for the
amount George earned.
b. Graph the equations in the system. Be sure to label all axes and show your scale.
c. Use your graph to determine how much each person earned.

Answers

Answer:

Step-by-step explanation:

g = s + 75

g + s = 525

That's just the answer for (a). You'll have to graph it yourself.

Guide to the methodology:

You need to plot 2 points for each equation and draw a line between the points. Where the two lines intersect gives you the values that solve the equations.

To get your co-ords,

Let g = 0

s = -75

Create a graph with g as one axis and s as the other. Plot a point of (0,-75)

Let s = 0

g = 75

Plot another point (0,75)

Draw a line between the two points. This is your line for g = s +75

You then need to plot a line for the second equation.

Let g = 0. S = 525. First co-ordinate is (0,525)

Let s = 0. G = 525. Second co-ordinate is (525,0)

Neli needs to cut 3 long boards into 9 smaller boards.The first is 10 ft, the second is 16 ft, and the third is 18 ft. The table lists the smaller boards Neli needs use a drawing to show how he can devide the 3 boards so there is no waste.

Answers

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Answers

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You're looking for the smallest number that's a multiple of both 30 and 40.
If you've learned about "Least Common Multiple" by now, you'll be happy
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That's the 4th multiple of 30 and the 3rd multiple of 40.

If the trains are both in the station right now, then the next time they'll
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How to solve it?

Answers

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Answers

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Answers

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