Logarithm equations: 64^x=8 and also log base 64 y= 1/3?

Question

Please show work on these 2 problems. Thank you.

Answer 1

1)

64^x = 8

Your first step is to realize that 64 is in actuality a power of 8; in fact, 64 is equal to 8 squared! Using this knowledge, we replace 64 with 8^2, giving us

[8^2]^x = 8

As one of the exponential rules, whenever we have a power to a power, we can multiply the exponents, giving us

8^(2x) = 8

Remember that 8 is equal to 8^1, so

8^(2x) = 8^(1)

And now, we have the same base. This means we can equate the exponents.

2x = 1, or x = 1/2

2) Log [base 64](y) = 1/3

Your first step is to convert this to logarithmic form. Remember that log [base b](a) = c converts to b^c = a. Thus,

64^(1/3) = y

And, it takes some knowledge and intuition, but 64 is actually a perfect cube, so we can take the cube root easily. It's 4.

y = 4

Answer 2

If 64^x = 8 then x = 1/2.
This is because the square root of 64 is 8, and square root is the same as an exponent of 1/2.

If log base 64 (y) = 1/3 then y = 64^(1/3). This is by converting logarithmic form to exponential form. So, y = 4. The 1/3 exponent means take the cube root, and the cube root of 64 is 4.

Answer 3

64^x=8

Get a common base of 2, so you get:
(2^6)^x=2^3
Then distribute out the exponents so you get:
2^6x=2^3
Set the exponents equal to each other:
6x=3
Then divide both sides by 6
x=3/6
Reduce the fraction to ½.


Log base 64y=1/3

64 is the base, 1/3 is the exponent and y is the answer , so set it up as:
64^1/3=y

Enter into a calculator as:
64^(1/3) or as the cube root of 64

because when an exponent is a fraction, the nominator is the power the number is raised to and the denominator is the index that goes in the radicand.

If entered correctly,
y=4

Answer 4

a) log(64) 8 = x [64 is the subcript]
x = 1/2

b) 64^(1/3) = y
y = 4

Answer 5

log(base a)b=c ability a(power c)=b or a raised to c equals b.for eg: log(base 10)a hundred=2 coz 10 raised to 2 delivers a hundred. RHS OF UR QN IS so log (base 4)2=a million/2 coz sq. root of four is two. LHS log(base 2)Y=a million/2 Y=2 raised to 0.5 ie sq. root of two =a million.414