log x = 1. the answer is 10 but i don't know how they got that. Please explain.
5^√(2x)=2^(x+2).
2006-12-28 06:47:59 · 8 answers · asked by Ha!! 2 in Science & Mathematics ➔ Mathematics
log x means "the power to which you would raise 10 in order to get x".
So, if you raise 10 to the first power, what do you get? 10, right?
You can also think about it as raising both sides to a power of 10. Then since log is the opposite of exponentiation, they cancel.
10^(log x) = 10^1
x = 10^1 = 10
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5^√(2x)=2^(x+2)
Take log of both sides:
log 5^√(2x) = log 2^(x+2)
Then by the power rule:
√(2x) log 5 = (x+2) log 2
√(2x) = log 2/log5 (x + 2)
Square both sides:
2x = (log 2/log 5)² (x² + 4x + 4)
(log 5/log 2)²2x = x² + 4x + 4
0 = x² + (4 - 2(log 5/log 2)²)x + 4
Which is messy, but you can plug it into the quadratic formula and get values for x. Don't forget to discard any negative answers, because they wouldn't work with the original equation.
2006-12-28 06:56:19 · answer #1 · answered by Jim Burnell 6 · 1⤊ 0⤋
Laws of Logs
log without a base number is always log base 10.
log(10) x = 1, therefore 10^1 = x, x = 10
Second one:
sqrt 2x log 5 = (x+2) log 2 square both sides
2x log 5 = (x+2)^2 log 2
(2x)(log 5/log 2) = (x+2)^2
Use a calculator to calculate log 5/log 2, then solve.
2006-12-28 14:59:06 · answer #2 · answered by p_rutherford2003 5 · 1⤊ 0⤋
The first one is straight forward . For log x = 1, you can read it as log x to base 10 is 1. In the equivalent exponent form, this is written 10^1 = x. Therefore, x = 10.
The second one is more difficult. Take the log of both sides to start.
root(2x) log 5 = (x + 2)log 2
Solve this equation.
2006-12-28 15:01:59 · answer #3 · answered by keely_66 3 · 1⤊ 0⤋
1) By definition, log function is the inverse of base 10 power function. Therefore,
10^(log x) = 10^1 , which is equivalent to
x = 10.
2) Let t =â (2x), t ⧠0
Solve for x,
x = t^2/2
The original equation becomes
5^t = 2^(t^2/2 + 2)
Take ln on both sides,
t ln5 = (t^2/2 + 2)ln2
which can be simplified to,
t^2 -( 2ln5/ln2)t + 4 = 0
Solving for t gives,
t = 1.14237 or 3.50148
Therefore, x = 0.65250 or 6.13019
2006-12-28 15:24:25 · answer #4 · answered by sahsjing 7 · 1⤊ 0⤋
log x = 1
The log shows what power of ten a number is. For example the log of 100 is 2 because 100 is 10^2. The log of 3.16 is .5 because 10^.5 (the square root of ten) is 3.16.
log x = 1 is the same as 10^1 = x, so x = 10
2006-12-28 14:59:50 · answer #5 · answered by Steve A 7 · 1⤊ 0⤋
This is an exercise in opposite opperations. Just as addition is the opposite to subtraction and division is the opposite of multiplication, you can cancel out the logarithm by rasing the base of the logarithm to itself. i.e. 10^log(x). Basic algebra says that this can only be done if applied to each side of the equation so the steps follow through and you're left with the simple solution.
log(x) = 1 -> 10^log(x) = 10^1 -> x = 10^1 -> x = 10
Hope that helps!
2006-12-28 15:00:42 · answer #6 · answered by Zac S 2 · 1⤊ 0⤋
You can rewrite the problem as 10 ^ 1 = X.
There is an assumed base of 10 whenever it is not given. If another base is given, it will be a subscript to the right of the word "log."
2006-12-28 15:00:22 · answer #7 · answered by puntjack 1 · 1⤊ 0⤋
you should consult my math teacher.. he would definitely know this one! I sure dont ... sry :(
2006-12-28 14:51:01 · answer #8 · answered by ? 2 · 0⤊ 2⤋