Someone! ANyone! Please help me solve these log's so i can study them thanks!?

Question

1) log (base)4 (x^3 + 37 ) = 3



2) lnx + lnx ^2= 3



3) ln ( 5+4x ) - ln ( 3x+x ) - ln3 = 0



4) f(x) = 4^x - 7

Answer 1

1)x^3+37 = 4^3
x^3+37 = 64
x^3 = 27
x=3

2)lnx +lnx^2 = 3
lnx+ 2lnx =3
3lnx = 3
lnx =1
x = e

3) and 4) seems to be incomplete

Answer 2

In many log and exponent problems, the trick is to have logs (of the same base) on both sides.

Remember that a log means: a number to which one raises the base. If the base is 4 and the number is 64, then the log(base4) of 64 is 3 (because 4^3 = 64)

So, in number 1, replace the 3 on the right side with:
log(base4) 64

log (base)4 (x^3 + 37 ) = 3
log (base)4 (x^3 + 37 ) = log(base4) 64

Once you have this, then if both sides are equal (if the two log values are equal), it must be because the numbers themselves -- before doing the log -- are equal:

x^3 + 37 = 64
x^3 = 27
x = 3

2 and 3) for logs that are indicated with ln (natural logs), the base is a value we call e.
e=2.718281828459045235...
The number itself is not important. What is important is that you put everything in natural logs.

ln (e^y) = y, whatever y is.
Remember the definition: a log is the power to which you raise the base to get the number. If the number is already expressed as the base raised to a power, then the log is that power.

Also, logarithms are a transformation of exponent calculation. when you multiply factors where a number is raised to different powers, you add the exponents:

(2^3)*(2^5) = 2^(3+5) = 2^8
If we did the operation in log(base2), we'd get:
log(base2)(2^3) + log(base2)(2^5) = log(base2)(2^8)
3 + 5 = 8

Combining these tricks:
2) ln(x) + ln(x^2) = 3
ln[(x)*(x^2)] = 3
ln(x^3) = 3
ln(x^3) = ln(e^3)
x^3 = e^3
therefore x must be the same as e.

There are many ways to do number 3 (and we should get the same answer, whatever method we choose).

Subtracting logs is the same as dividing the numbers.

For example: (2^8) / (2^3) = 2^5
log(base2)(2^8) - log(base2)(2^3) = log(base2)(2^5)
8 - 3 = 5

One way is to get rid of the zero by moving something from the left to the right:
ln ( 5+4x ) - ln ( 3x+x ) - ln3 = 0
ln(5+4x) = ln(3x+x) + ln(3)
ln(5+4x) = ln[3(3x+x)] = ln(12x)
Therefore,
5 + 4x = 12x
5 = 12x - 4x = 8x
5/8 = x

-----


(all bases must be the same within the same equation)

If Log(A) = Log(B), then A+B
If Log(A) + Log(C) = Log(B), then (A*C) = B
If Log(A) - Log(D) = Log (B), then (A/D) = B
If X*Log(A) = Log(B), then (A^X) = B

Similarly:

If X^A = X^B, then A=B
If (X^A)*(X^C) = X^B, then (A+C) = B

---

You must reduce each side to one log expression. For example, in number 2, we had on the left:
ln(x) + ln(x^2)
we had to use the transformation of "addition of logs" to "log of a multiplication" to get
ln[(x)*(x^2)]
in order to have only one expression on the left.

In number 3, we did that by keeping only one on the left and moving the other two to the right (where we could use the same tranformation to turn the addition of logs into the log of the multiplication).

Answer 3

1) log (base)4 (x^3 + 37 ) = 3

Rewrite this one as.....
4^3 = x^3 + 37

Now isolate x.
4^3 - 37 = x^3
64 - 37 = x^3
27 = x^3

Now take the cube root of each side.

3 = x

That's the solution WITH an explanation.

hope that helps! :-)