Please explain how you reached to the answer. Answers are in brackets. Many thanks!
1) 8(2x - 1)^3 = 125 [ANS:7/4]
2) 2^(x-1) - 2^x = 2^-3 [ANS: no solution]
3) 3^(x+1) + 3^x = 36 [ANS: 2]
4) log(base x) 16 = 4/3 [ANS: 8]
5) log(base 9) 3 times sqroot of 3 = x [ANS: 3/4]
1) 8(2x - 1)^3 = 125 [ANS:7/4]
2) 2^(x-1) - 2^x = 2^-3 [ANS: no solution]
3) 3^(x+1) + 3^x = 36 [ANS: 2]
4) log(base x) 16 = 4/3 [ANS: 8]
5) log(base 9) 3 times sqroot of 3 = x [ANS: 3/4]
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You really don't need logarithms for the first, although I suppose you could use them. Divide both sides by 8 and noticethat the new right hand side, 125/8, is 5^3/2^3 = (5/2)^3, so if you take cube roots of both sides you conclude that 2x - 1 = 5/2 and hence 2x = 5/2 + 1 = 7/2 and hence x = 7/4.
In (2) I would note tha 2^(x - 1) = (2^x)/(2^1) = (1/2) 2^x, so the equation is (1/2) 2^x - 2^x = 2^(-3). Since (1/2) a - a = (1/2) a - 1 a = (1/2 - 1) a = (-1/2) a for any a, we see that the left hand side of this is (-1/2) 2^x and hence we are to solve (-1/2) 2^x = 2^(-3). Multiplying both sides by -2 we see that we need 2^x = -2^(-2) = -1/4 and this has no solutions because 2^x is always positive no matter what x is.
In (3) I would note that 3^(x + 1) = 3^x 3^1 = 3 * 3^x, so the left hand side is 3 * 3^x + 3^x = 4 * 3^x, so we are to solve 4 * 3^x = 36. Dividing both sides by 4 we see that 3^x = 9 and we could use logarithms here to solve for x, or just note that 9 = 3^2 so clearly this is solved by x = 2.
I will write log_b(c) for the base b logarithm of c. The fundamental property of this number is that when you raise b to it, you get c, ie, b^(log_b(c)) = c. So, in (4), we conclude that x^(the left hand side) = x^(the right hand side) and this is the statement that 16 = x^(4/3). Raising both sides of this equation to the 3/4 power we find that 16^(3/4) = x, and since 16 = 2^4 we see that 16^(3/4) = (2^4)^(3/4) = 2^(4 * 3/4) = 2^3 = 8, so x = 8.
In (5) sinc 9^(left hand side) = 9^(right hand side) we deduce that 3 * sqrt(3) = 9^x. Since the left hand side is 3 * 3^(1/2) = 3^(1 + 1/2) = 3^(3/2) and the right hand side is 9^x = (3^2)^x = 3^(2x) we se that 3^(3/2) = 3^(2x) so that 2x = 3/4 and hence that x = 3/4.
Alternatively we could note that this problem is just asking us to evaluate the base-9 logarithm of 3 sqrt(3). The simplest way to do this is just to write 3 sqrt(3) as a power of 9; whatever power of 9 we need to use in order to do this, is the answer. Now we have 3 * sqrt(3) = 3 * 3^(1/2) = 3^1 * 3^(1/2) = 3^(1 + 1/2) = 3^(3/2) and since 3 = 9^(1/2) we see that 3 * sqrt(3) = (9^(1/2))^(3/2) = 9^(1/2 * 3/2) = 9^(3/4), so the answer is again 3/4.
In (2) I would note tha 2^(x - 1) = (2^x)/(2^1) = (1/2) 2^x, so the equation is (1/2) 2^x - 2^x = 2^(-3). Since (1/2) a - a = (1/2) a - 1 a = (1/2 - 1) a = (-1/2) a for any a, we see that the left hand side of this is (-1/2) 2^x and hence we are to solve (-1/2) 2^x = 2^(-3). Multiplying both sides by -2 we see that we need 2^x = -2^(-2) = -1/4 and this has no solutions because 2^x is always positive no matter what x is.
In (3) I would note that 3^(x + 1) = 3^x 3^1 = 3 * 3^x, so the left hand side is 3 * 3^x + 3^x = 4 * 3^x, so we are to solve 4 * 3^x = 36. Dividing both sides by 4 we see that 3^x = 9 and we could use logarithms here to solve for x, or just note that 9 = 3^2 so clearly this is solved by x = 2.
I will write log_b(c) for the base b logarithm of c. The fundamental property of this number is that when you raise b to it, you get c, ie, b^(log_b(c)) = c. So, in (4), we conclude that x^(the left hand side) = x^(the right hand side) and this is the statement that 16 = x^(4/3). Raising both sides of this equation to the 3/4 power we find that 16^(3/4) = x, and since 16 = 2^4 we see that 16^(3/4) = (2^4)^(3/4) = 2^(4 * 3/4) = 2^3 = 8, so x = 8.
In (5) sinc 9^(left hand side) = 9^(right hand side) we deduce that 3 * sqrt(3) = 9^x. Since the left hand side is 3 * 3^(1/2) = 3^(1 + 1/2) = 3^(3/2) and the right hand side is 9^x = (3^2)^x = 3^(2x) we se that 3^(3/2) = 3^(2x) so that 2x = 3/4 and hence that x = 3/4.
Alternatively we could note that this problem is just asking us to evaluate the base-9 logarithm of 3 sqrt(3). The simplest way to do this is just to write 3 sqrt(3) as a power of 9; whatever power of 9 we need to use in order to do this, is the answer. Now we have 3 * sqrt(3) = 3 * 3^(1/2) = 3^1 * 3^(1/2) = 3^(1 + 1/2) = 3^(3/2) and since 3 = 9^(1/2) we see that 3 * sqrt(3) = (9^(1/2))^(3/2) = 9^(1/2 * 3/2) = 9^(3/4), so the answer is again 3/4.