SOLUTION: If log 16 64 =2 x + 1, what is the value of x ?

Question 921991: If log 16 64 =2 x + 1, what is the value of x ?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
log16(64) = 2x + 1

this is true if and only if 16^(2x+1) = 64

you could solve this by taking the log10 of both sides of the equation and you will find the answer.

i'll do that after i show you another way, if you recognize it.

16 = 4^2, so 16^(2x+1) = (4^2)^(2x+1) = 4^(4x+2)

16^(2x+1) = 64 is equivalent to 4^(4x+2) = 64

if you know that 64 = 4^3, then you can make your equation become:

4^(4x+2) = 4^3

this is true if and only if 4x + 2 = 3
solve for x to get x = 1/4

your original equation becomes:

log16(64) = 2(1/4) + 1) which becomes:

log16(64) = 3/2.

this is true if and only if 16^(3/2) = 64

16^(3/2) is equal to (16^(1/2))^3 which is equal to 4^3 which is equal to 64, so your value of x is good.

if you did not recognize all of the above, then you could have solved this problem as follows:

log16(64) = 2x + 1 if and only if 16^(2x+1) = 64

take the log10 of both sides of this equation to get:

log(16^(2x+1) = log(64)

since log10 is normally just shown as log, there's no loss of accuracy here.

since log(16^(2x+1) = (2x+1) * log(16), this equation becomes:

(2x + 1) * log(16) = log(64)

divide both sides of this equation by log(16) to get:

2x + 1 = log(64) / log(16)

use your calculator log function to get:

2x + 1 = 1.5

subtract 1 from both sides of this eqaution to get:

2x = .5

divide both sides of this eqaution by 2 to get:

x = .5 / 2 which makes x = 1/4.

your could also have solved this using the log conversion formula of:

log16(64) is equivalent to log10(64)/log10(16)

log10(64)/log10(16) = 2x + 1

since log10 is normally just shown as log, you get:

log(64) / log(16) = 2x + 1 which is the same as:

2x + 1 = log(64) / log(16)

that's the same equation we got above when we took the log of both sides of the equation, so the answer will be the same.

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