Help with changing log bases?

Question

I already asked this question, but I did not recieve a clear enough answer, and they changed base 2 to 4.

log base 2 (3x+2) - log base 4 (X) = 3

The book says to change log base 4 to 2. Its not so much the answer I am looking for, just how this works.

Answer 1

ex: If you want to change the base from b to c of log(a)

log a to the base b = [log a / log b] to the base c

similarly

log(x) [to the base 4] = log(x)/log(4) [to the base 2]

log[4](x) = log[2](x)/log[2](2^2)

log[4]x = log[2](x)/2log[2](2) (since loga^2 = 2 log(a))


log[4](x) = log[2](x)/2 (log of any number to the base of same

number is 1)

Answer 2

Your problem coincidentally (or maybe by author's design) has the latter base being the square of the first base.

You need to know that a logarithmic form has its equivalent exponential form:

If Log (base B) of (ARG) = X

then B^X = ARG [Were the bases in your problem not so nicely related, then we would be taking the NATURAL (base e) or COMMON (base 10) log of each side of this last equation, getting

X ln B = ln ARG; X = [ln ARG] / [ln B] *** to change from base B to base e.

However, in your problem, second term... for log base 4 of x (let it equal z) we have 4^z = x, but since 4 = 2^2 then

2^(2z) = x. Take log (base 2) of each side and get

2z = log (base 2) x
z = [log (base 2) x ] / 2 [which is the same as the *** formula above dictates]

So your original prob becomes
log(base 2) of [3x + 2] - [log(base 2) of x] /2 = 3

Simplifying (multiply thru by 2) gives

2 [log(base 2) of [3x + 2] - [log(base 2) of x] ] = 6

log(base 2) of { ( [3x+2]^2) / x } = 6

or { ( [ 3x+2]^2 ) / x } = 2^6 = 64 which becomes a quadratic equation:

9x^2 - 52x + 4 = 0 that I was loathe to solve.
---------------
Finally, after a half-way good night's sleep your x comes out

x = [26 plus or minus 8*sqrt(10) ] / 9
which produces these approximations:

x = 5.6998 or .07798 [and my cking these in your original equation seems to validate both]

Answer 3

There is a math formula you need. The formula is something like Log1000 / Log100 = Log 10. I don't remember too well what's the formula. Google it, and apply it.

Answer 4

log base2 (3x+2)-log base2 (2x) = 3

4=2^2 so move the exponent 2 and multiply it to x.

Answer 5

since 4 is 2^2; log_4 (x) = log_2 (X^2)
log_2(8)= 3