I need alot of help with Logarithimic functions?

Question

1) Log 2 64
2) log 2 square root of 2
Convert to a logarithmic equation
1) 5^-3= 1/125
2) 10^0.3010=2
And convert to an exponential equation
1) t=log 4 7
2) log 7=0.845

Any help will be much appreciated. Thanks

And log64 4

Answer 1

1) The definition of a logarithm says : if a^X = N, then X = log[a]N.
Turn this around to say : if X = log[a]N, then a^X = N.
Now you can see what to do if we let X = log[2]64.
We get : 2^X = 64 = 2^6.
Therefore, X = 6, that is, log[2]64 = 6.

2) Same with this one. Let X = log[2]sqrt(2).
Thus, 2^X = sqrt(2) = 2^(1 / 2).
Therefore, X = 1 / 2.

3) 5^(-3) = 1 / 125
Use the definition again. If a^X = N, then X = log[a]N.
Here we have : a = 5, X = -3 and N = 1 / 125.
Thus, -3 = log[5](1 / 125)

4) 10^(0.3010) = 2
In the same way, let a = 10, X = 0.3010 and N = 2.
Thus, 0.3010 = log[10]2.

5) t = log[4]7
Definition again : 4^t = 7.

6) log[10]7 = 0.845
Definition again : 10^(0.845) = 7.

7) Let X = log[64]4
By the definiton : 64^X = 4.
Or, (2^6)^X = 4.
Or, 2^(6X) = 2^2.
Therefore, 6X = 2, or, X = 1 / 3.

Answer 2

log 64 to base 4 is 3, since 4^3 = 64*
log 64 to base 2 is 6, since 2^6 = 64*
2 raised to the power half (1/2 or 0.5) = sqrt 2. Hence answer is 1/2 or 0.5*
1/125 =125^-1 =5^-3. Thus, R.H.S = L.H.S.*
10^0 = 1 (question NOT understood)
e^t = 47, may be wrong.
7 = e^0.845, may be wrong.

*Friend, I could be wrong in some (or even all) of these answers, as I have left Maths, more than 35 years ago. Please seek corrections from your teacher. In, t = log 4 7, I could not understand the gap between 4 and 7.

Answer 3

1) 64 = 2^6 log(base 2) 64 = 6
2) square root of 2 = 2^(1/2) log(base 2) square root of 2 = 1/2

convert to logarithmic eq:
1) 5^(-3) = 1/125 log(base 5) 1/125 = -3
2) 10^0.03010 = 2 log 2 = 0.03010

convert to exponential eq:
1) 4^t = 7
2) 10^0.845 = 7

not really sure what you mean by log 64 4

if you're talking about: log(base 4) 64 = 3
if you're talking about: log(base 64) 4 = 1/3

Answer 4

1) log 2 64 can be written as log 2 2^6 which is 6 log 2 2 which is 6
2) log 2 sqrt(2) is (1/2)log 2 2 which is 1/2

To convert to logarithm take log on both sides

To convert to exponential take antilog on both sides.

Answer 5

1) 6
2) 1/2
---------------------
1) log 5 1/125=-3
2) log 10 2 = 0.3010
---------------------
1)4^t =7
2)10^(0.845)=7