Logs - Solve 2^(x+4) = 3^(x-1)?

Question

Just incase you didnt understand, ^ means that it is to the power of, so 2^(x+4) means 2 to the power of x+4. Any help would be appreciated.

(can i also have an explanation as to where the answer came from, as 7 doesnt seem to work)

Answer 1

7

Answer 2

7

Answer 3

Hello!

To start with, you are right and 7 is not the answer to the question.

You want to solve for x.
We have 2^(x+4) = 3^(x-1)

Take logs on both sides...
log(2^(x+4)) = log(3^(x-1)

Recall the property of the logarithms:
log(a^b) = b*log(a).
Using this, we get:
(x+4)*log(2) = (x-1)*log(3)

Expanding the above equation gives:
x*log(2) +4*log(2) = x*log(3) - log(3)
x*(log(2) - log(3)) = - log(3) - 4*log(2)
- x*(log(2) - log(3)) = log(3) + 4*log(2)
x*(log(3) - log(2)) = log(3) + log(2^4)
x*(log(3) - log(2)) = log(3) + log(16)

Now, recall another property of the logarithms:
log(a) + log(b) = log(ab)
log(a) - log(b) = log(a/b)
Using this property the equation becomes:
x*(log(3/2)) = log(3*16)
x = log(48)/log(1.5) - this is the exact answer and cannot be simplified.

Use your calculator to obtain an approximate solution. That is:
x = (3.871201011)/(0.405465108)
x = 9.547556457
x = 9.55 (3 significant figures)

Note: There is no need to provide the approximate solution, unless the question asks for it.

Hope this helps.
Thank you.

GOOD LUCK

Answer 4

2^(x+4) = 3^(x-1)

Take logs of both sides
(x+4) lg 2 = (x-1) lg 3

Expand
x lg 2 + 4 lg 2 = x lg 3 - lg 3

Rearrange
4 lg 2 + lg 3 = x(lg 3 - lg 2)

Use log rules to simplify
lg 2^4 + lg 3 = x lg (3/2)
lg 16 + lg 3 = x lg (3/2)
lg 48 = x lg (3/2)

x = (lg 48)/[lg (3/2)]

That is the exact answer, an approximate numerical answer is 9.5476

Answer 5

here's a simple way:-

2^(x+4) = 3^(x-1)
use rules of indices
2^x*2^4=3^x*3^(-1)
16*2^x=3^x*(1/3)
48*2^x=3^x
take natural logs of each side
ln48+xln2=xln3
x(ln3-ln2)=ln48
>>>x=ln48/(ln3-ln2)
=9.547556457
{by T-80 calculator}

what we are saying is that

2^(13.547556457)
=3^(8.547556457)

i hope that this helps

Answer 6

log 2^(X + 4) = log 3^(X - 1)
2^X x 2^4 = 3^X x 3^(-1)
16 x 2^X = 1/3 x 3^X
48 x 2^X = 3^X
48 = (3^X) / (2^X)
log 48 = log[(3^X) / (2^X)] = log 3^X - log 2^X
log 48 = X log 3 - X log 2 = X log(3/2)
X = log 48 / log(3/2)
X = 9.548

Answer 7

2^(x + 4) = 3^(x - 1)

Take the log of both sides:
(x + 4)log(2) = (x - 1)log(3)

Expand:
xlog(2) + 4log(2) = xlog(3) - log(3)

Put x's on one side and constants on the other:
xlog(3) - xlog(2) = 4log(2) + log(3)

Take out factor x from LHS:
x[log(3) - log(2)] = 4log(2) + log(3)

Now the LHS is xlog(3/2)

and the RHS is log(2^4) + log(3) = log(2^4 * 3) = log(48)

Therefore, x log(3/2) = log(48)

Thus, x = log(48) / log(3/2) = 9.5475565 (to 8 significant digits)

Answer 8

I'm going with 7. Final Answer.

Answer 9

2^(x + 4) = 3^(x - 1)

Your first step is to take the log of both sides. I'm going to take log[base 2].

log[base 2](2^(x + 4)] = log[base 2](3^(x - 1))

By the log property of log[base 2](a^c) = c*log[base 2](a), we have

(x + 4)log[base 2](2) = (x - 1) log[base 2](3)

Note that log[base 2](2) = 1, so

x + 4 = (x - 1) log[base 2](3)

Expand the right hand side.

x + 4 = xlog[base 2](3) - log[base 2](3)

Move xlog[base 2](3) to the left hand side, and the +4 to the right hand side.

x - xlog[base 2](3) = -4 - log[base 2](3)

Factor x.

x(1 - log[base 2](3) = -4 - log[base 2](3)

Divide accordingly.

x = [-4 - log[base 2](3)] / [1 - log[base 2](3)]

If you want a calculator friendly answer, then by the change of base formula, log[base 2](3) = ln(3) / ln(2), so

x = [-4 - ln(3)/ln(2)] / [1 - ln(3)/ln(2)]

Multiply top and bottom by ln(2)

x = [-4ln(2) - ln(3)] / [ln(2) - ln(3)]

x = [ln(2^(-4)) - ln(3)] / [ln(2) - ln(3)]

x = [ln(1/16) - ln(3)] / [ln(2) - ln(3)]

By the log subraction property ln(a) - ln(b) = ln(a/b),

x = [ln(1/48)] / [ln(2/3)]

Answer 10

2^(x+4) = 3^(x-1)
Taking logs of both sides, we have (x+4)log2 = (x-1)log3
Therefore (x+4)/(x-1) = log3/ log2 = 1.584962501
Therefore x+4 = 1.585*(x-1) { rounding up }
Therefore 5.585 = 0.585x or x = 5.585/0.585 = 9.547