Topic: Logarithms Question 10 points will be awarded to best answer?**Question:**
Simplify:
log4 32 - log9 27
1 + 2log10 (x + 1)= log10(2x + 1) + log10 (5x + 8)
If you choose to answer these 2 questions can you please show the working out along with your answer please so i can learn. 10 points will be awarded to best answer and i appreciate any help i get with these questions.

April 21, 2019 / By Topaz

The following seven theorems are the most important basic theorems that you will ever know. If you know them and can use them your teacher cannot write a log test on which you won't get an A unless you simply don't have the time to finish the test. I have taught math and logarithms enough to guarantee that. Theorem 1) The base of any logarithm must be a positive real number. Theorem 2) log(base a) of x = log(x) / log(a) where the last two logs can be of any base. (This may come up, but if you're going to skip any of these this is the one to skip). Theorem 3) log(d^c) = c*log(d) for every positive real number d, for every real number c, and every base possible. Theorem 4) log(x * y) = log(x) + log(y) for any base and for both x and y positive. The logarithm of z is not defined for z <= 0 Theorem 5) log(x / y) = log(x) - log(y) for any base and for both x and y positive. The logarithm of z is not defined for z <= 0 Theorem 6) For any base b and any positive real number x, b^[log(baseb) (x)] = x Theorem 7) For any base b and any real number x, log(baseb) (b^x) = x. I will be referring to these theorems by number throughout the solutions to your problem. (1) will mean Theorem 1) and so on. ------------------------ 1) log4 32 - log9 27 First log4 (32) Since 2 is the square root of 4 and since 32 is a power of 2, we are interested in the fact that 2^5 = 32. 1) 2^5 = 32 2 = sqrt(4) = 4^(1/2) 3) 2 = 4^(1/2) we can raise both sides of this equation to the fifth power, getting 2^5 = (4^(1/2))^5 (4(1/2))^5 = 4^(5/2) (This is due to an exponent law: ((x^a)^b) = (x^(a * b)) By Theorem (7) log4 (4^5/2) = (5/2)...<<<...Answer to #1 - part 1 Now for log9 (27) This is just like what we did above. sqrt(9) = 9^(1/2) = 3 27 = 3^3 and: 3 = 9^(1/2) Raise both of these to the power 3. 3^3 = (9^(1/2))^3 and by the exponent law (9^(1/2))^3 = (9)^(3/2) So 27 = 9^(3/2) and log9 (27) = log9 (9^(3/2) and using (7) again log9 (9^3/2) = 3/2 ..<<..Part 2 Since the answer to the first part was (5/2) So log4 32 - log9 27 = (5/2) - (3/2) = (2/2) = 1...<<<...Answer to Problem 1 ------------------ 2) 1 + 2log10 (x + 1)= log10(2x + 1) + log10 (5x + 8) Using (7) again, 1 = log10 (10). Now the equation is all logs and it is: log10 (10) + 2log10 (x + 1)= log10(2x + 1) + log10 (5x + 8) Using (3) 2 * log10 (x + 1) = log10 ((x+1)^2) *********** This next theorem is used so frequently that I'm going to copy it here: Theorem 4) log(x * y) = log(x) + log(y) for any base and for both x and y positive. The logarithm of z is not defined for z <= 0 *********** So: log10 (10) + log10 ((x+1)^2)= log10(2x + 1) + log10 (5x + 8) is the same as: log10 (10 * (x^2 + 2x + 1)) = log10 (10x^2 + 21x + 8) Since the logs of these two expressions are equal, they are equal, so: 10x^2 + 20x + 10 = 10x^2 + 21x + 8..........Subtract the 10x^2 terms from both sides of this equation leaving: 20x + 10 = 21x + 8..........Subtract 8 from both sides 20x + 2 = 21x......Subtract 20x from both sides 2 = x x = 2.....<<<<<.....Answer to Problem 2 I hope that this helps you in your course. I always had my students copy the seven Theorems down and keep them by their books when they were doing logarithm homework. They're all in every algebra textbook that I have ever seen, but they're scattered over many pages. I advise copying them down and doing what my students did. Good luck :D .

👍 194 | 👎 3

Did you like the answer? Do some research and realize what is and isn't myth. You can't control a reptile's size by attempting to stuff it into a smaller cage. You can make it ill, cause it stress, and it can even injure itself trying to move in a too small tank, or give up moving, and develop other health issues. Reptiles grow for their entire lives. The only way to stunt its growth is by abuse or inadequate nutrition. Keep in mind that: " Bearded Dragons live about 8–15 years with proper care in captivity, though some can live up to 20 years old Bearded Dragon have broad triangular heads and flattened bodies, with adults reaching approximately 18 to 24 inches (45-60 cm) head including tail. " http://en.wikipedia.org/wiki/Bearded_Dra... So, why would you want to attempt to stuff something that can very well live with you for 20 yr., and get 2 ft. long into a tank only 3 ft.long x 12" wide? This would be the equivilent of you living in your bathtub for your whole life. You could lay down and turn around, but how happy would you be? The minimum suggested size tank for a healthy and happy beardie is a 40 gal. reptile/breeder tank. Babies should be kept on paper towels, and adults should be kept on slightly textured ceramic tile (not smooth or gloss). http://hubpages.com/hub/Impaction http://hubpages.com/hub/Fake-Rock-Instru... (note tile floor) Have you done the rest of your homework? UVB: http://www.uvguide.co.uk/ http://www.anapsid.org/maincaptive.html Reptile vitamins and calcium powder with D3 is needed. MBD: http://www.dachiu.com/care/abeard.html http://hubpages.com/hub/Metabolic_Bone_Disease Please don't buy an animal you can't provide for properly. It's not fair to the pet. If you don't have the room for a larger tank, then select a smaller lizard. Good luck, and I hope this has been helpful.

32 = 4^2.5 so log(base 4)32 = 2.5. Likewise, log (base9) 27 = 1.5. That's what logs mean. 1 + 2log10 (x + 1)= log10(2x + 1) + log10 (5x + 8) Use 1 = log10 10. This gives log10 10 + 2log10 (x + 1)= log10(2x + 1) + log10 (5x + 8) Then use log a + log b = log (ab) to get 10(x + 1)^2 = (2x + 1)(5x + 8) The rest is simple algebra

👍 80 | 👎 -5

a) log4 32 - log9 27 ---> log2^2 2^5 - log3^2 3^3 ---> 5/2log2 2 - 3/2 log3 3 ---> 5/2 - 3/2 = 1 b) 1 + 2log10 (x + 1)= log10(2x + 1) + log10 (5x + 8) ---> log10 10 + log10 (x+1)^2 = log10 [(2x+1)(5x+8)] ---> log10 10.(x+1)^2 = log10 [(2x+1)(5x+8)] ----> 10(x^2+2x+1) = (2x+1)(5x+8) Solving the above equation results in x =2. - [email protected]!m

👍 78 | 👎 -13

32=4^2.5, 27=9^1.5, 2.5-1.5=1 ====================================== 1=log10 10(x+1)2=(2x+1)(5x+8) 10x^2+20x+10=10x^2+16x+5x+8 2=x God bless you.

👍 76 | 👎 -21

im assuming the bottom to be 10 right here is going log 5x = three 5x = 10^three x=one thousand/five = two hundred log x = a million/10 = 10^-a million = so x = -a million 10^2x = 4x10 taking go browsing either side 2x = log four x 10= log four + log 10 2x = log four +a million 2x = 2log2 +a million x = log2 +a million/two log(7x+three) = two so 7x+three = 10^two 7x+three=one hundred 7x = ninety seven x = ninety seven/7 2logx- log five = four log (x^two/five)=four x^two/five= 10,000 x^two=50,000 x= ?50,000 log(b5)(3x+10/four^three) = two (3x+10)/sixty four = 25 3x +10 = 25x64 3x+10 = 1600 3x= 1590 x = 530

👍 74 | 👎 -29

To convert Fahrenheit to Celsius (Centigrade), subtract 32 and divide by 1.8. To convert Celsius (Centigrade) to Fahrenheit, multiply by 1.8 and add 32.

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