Question 303567: log(7)x=5 the answer I got is x=16,954 and also log x =1 the answer I got is x=10 log(4) (x-3) = log(4) (x-3)=1 the answer I got is x=3 just not sure if I am right on these problems...I am having hard time understanding this.....and one I have no clue how to do is e^t=1000 any help would be greatly appreciated....thanks in advance
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! I presume you are saying the following:
log of x to the base 7 = 5
That would be written as log(7,x) = 5
By the basic definitions of logarithms, that can only be true if:
7^5 = x
Since 7^5 = 16807, then your answer has to be that x is equal to 16807.
Your answer should be (if I interpreted the problem correctly), that:
log(7,16807) = 5.
You can use your calculator to confirm that directly by using the log base conversion formula.
Using that formula:
log(7,16807) = log(10,16807)/log(10,7)
Solving that gets you:
log(7,16807) = 5, confirming that the answer is good.
Your second problem is log(x) = 1
Without specifying the base, the base of 10 is implied.
Your second problem equation becomes log(10,x) = 1
By the basic definition of logarithms, this is true if and only if:
10^1 = x
Since 10^1 = 10, your answer should be that x = 10.
your equation becomes log(10,10) = 1
This can written as log(10) = 1
Use your calculator to get log(10) and you'll see that it is equal to 1.
Looks like you got that one right.
Your third problem looks like log(4,(x-3)) = 1.
By the basic definition of logarithms, this is true if and only if:
4^1 = x-3 which becomes:
4 = x-3
Add 3 to both sides of this equation to get:
x = 7
Your answer should be x = 7.
Your equation becomes:
log(4,(7-3)) = 1 which becomes:
log(4,4) = 1
Use the base conversion formula to get:
log(4,4) = log(10,4)/log(10,4) which becomes:
log(4,4) = 1, confirming that the answer is good if I interpreted the problem correctly.
Your last problem is e^t = 1000.
To solve this, you need to take the log of both sides of this equation to get:
log(e^t) = log(1000).
The base of 10 is implied.
The formula could also be written as log(10,e^t) = log(10,1000).
Since, in general, log(x^y) = y*log(x), your equation becomes:
t*log(e) = log(1000).
Divide both sides of this equation by log(e) to get:
t = log(1000)/log(e).
e is equal to the scientific constant of 2.718281828.
Use your calculator to solve for t to get:
t = 6.907755279.
Substitute in your original equation to get:
e^6.907755279 = 1000.
Use your calculator to solve to get:
1000 = 1000, confirming that the answer is good.
The formulas that you needed to use to solve these problems are:
y = log(b,x) if and only if b^y = x.
That's the basic definition of logarithms.
Also:
log(x^y) = y*log(x).
Also:
log(b,x) = log(c,x) / log(c,b)
That's the base conversion formula.
I think I interpreted the problem correctly, but I may have gotten it wrong.
Terminology:
log(b,x) means log of x to the base of b.
log(x) means log of x to the base of 10.
Your calculator LOG function solves for logs to the base of 10.
Your calculator LN function solves for logs to the base of e.
e is the scientific constant of 2.718281828.
The basic definition of logarithms as it applies to the base of e is as follows:
log(b,x) = y if and only if b^y = x
Applied to a base of e, that becomes:
log(e,x) = y if and only if e^y = x
In your last problem, you could also have used the basic definition of logs to solve it.
That problem was e^t = 1000
By the basic definition of logs, that is true if and only if:
ln(1000) = t
Note that ln(1000) is the same as log(e,1000).
Since your calculator has an LN function key (or should have), you could have then used your calculator to solve this.
You would have gotten:
ln(1000) = 6.907755279
That's exactly the same answer we got before, as it should be.
You may be interested in my lessons on logarithms and exponents.
They can be found at the following links:
http://www.algebra.com/algebra/homework/logarithm/L.lesson
http://www.algebra.com/algebra/homework/logarithm/CTBOAL.lesson
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