SOLUTION: I had asked two problems a few days ago and have had no response here they are again 1. If possible evaluate g(t)for the given values of t g(t)=2t^3 - t^2 + 4 2. log2 2^6 I

Algebra ->  Finance -> SOLUTION: I had asked two problems a few days ago and have had no response here they are again 1. If possible evaluate g(t)for the given values of t g(t)=2t^3 - t^2 + 4 2. log2 2^6 I       Log On


   

Question 483270: I had asked two problems a few days ago and have had no response here they are again
1. If possible evaluate g(t)for the given values of t
g(t)=2t^3 - t^2 + 4
2. log2 2^6
I need help with these and they need detailed answers please help
Found 2 solutions by Theo, bucky:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
1. If possible evaluate g(t)for the given values of t
g(t)=2t^3 - t^2 + 4

you did not give any values of t so i wouldn't know how to answer question 1.
the procedure, however, would be the same as shown below:

equation g(t) = 2t^3 - t^2 + 4
solve for t = 3
the equation becomes:
g(3) = 2*(3)^3 - (3)^2 + 4
all you're doing is replacing t with 3 in the equation.
the answer, in that case, would be:
g(3) = 2*27 - 9 + 4 which becomes:
g(3) = 54 - 5 which becomes:
g(3) = 49


2. log(2,2^6)

you are looking to find the log of 2^6 to the base of 2.
since, in general, log(a^b) = b*log(a), your expression becomes:
6 * log(2,2)

to find 6 * log(2,2), you can use your calculator by converting the base of 2 to the base of 10 using the logarithm base conversion formula shown below.

6 * log(2,2) = 6 * ((log(10,2) / log(10,2))

the general formula is:

log(a,b) = log(c,b) / log(c,a)
a is the base you want to convert from.
c is the base you want to convert to.
b is the log value you are looking to solve for.
the expression of log of b to the base a is converted to:
log of b to the base c divided by log of a to the base c.

this allows you to use the LOG function of your calculator.

the formula becomes:

6 * log(2,2) = 6 * (LOG(2) / LOG(2)).

the answer is 6 * 1 because anything divided by itself is equal to 1.

your final answer is 6.

you would want to confirm your answer to make sure it's correct.
the basic definition of logs states:
y = log(b,x) if and only if b^y = x
your original expression is:
log(2,2^6)
set y equal to this to get:
y = log(2,2^6)
set x equal to 2^6 to get:
y = log(2,x)
set b equal to 2 to get:
y = log(b,x)
the basic definition of logarithms states that:
y = log(b,x) if and only if b^y = x
now, you know that b is equal to 2 and you know that x = 2^6, so this equation becomes:
2^y = 2^6
this equation is true if y = 6 because then you get:
2^6 = 2^6 which is true.
6 is the answer to your question.
log(2,2^6) = 6

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Your first problem cannot be done because you did not provide any values for t.
.
But to help you understand this problem, the problem is asking you to find the value of g(t) for a given value of t. For example, suppose the problem is to evaluate g(t) given that t equals 5. You do that by substituting 5 into the equation for g(t) everywhere that a "t" appears. The work would be done as follows:
.
Problem:
.
Evaluate g%28t%29+=+2t%5E3-t%5E2+%2B+4 when t+=+5
.
Begin by substituting 5 for every t in the expression
.
g%285%29+=+2%285%5E3%29-5%5E2%2B4
.
First recall that 5%5E3 is 5%2A5%2A5+=+125. Then recall that 5%5E2 is 5%2A5+=+25. Substitute these values into their places in the equation to get that:
.
g%285%29+=+2%28125%29+-+25+%2B+4
.
Multiply 2 times 125:
.
g%285%29+=+250+-+25+%2B+4
.
and simplify by subtracting 25 from 250 and then adding the 4:
.
g%285%29+=+225+%2B+4
.
and finally
.
g%285%29+=+229
.
This is the answer to this example problem. g(t) has been evaluated for t = 5.
.
Use this example as a pattern of how to work the problem for the values of t that you were given but did not provide to us.
.
Your second problem is to find:
.
log%282%2C2%5E6%29
.
First, by a rule of logarithms the exponent (in this problem the 6) can be brought out as a multiplier of the log. When you do this your problem then becomes:
.
6%2Alog%282%2C2%29
.
And by another rule of logarithms, when you are finding the logarithm of a quantity and the base that you are working in is the same as the quantity, the answer is always 1. As examples:
.
log%2810%2C10%29+=+1 and log%28e%2Ce%29=1 and log%2816%2C16%29=1
.
So, for your problem log%282%2C2%29+=+1 and you can substitute 1 for log%282%2C2%29 and this further reduces your problem to:
.
6%2Alog%282%2C2%29+=+6%2A1
.
So the answer to your problem is:
.
log%282%2C2%5E6%29+=+6
.
Hope this helps you. Good luck with your journey through mathematics.