Solve for x: 4^(x+2)=6^(2x-9)

a. 8.6
b. 24
c. 9

Thank you!

Take the logarithm of both sides:

log 4^(x+2) = log 6^(2x-9)

Use the property that log (a^b) = b log a:

(x+2) log 4 = (2x-9) log 6

Note that this will work no matter what base log you use as long as they are the same on both sides, so you could use base 10 logs or base e (natural logs). I will use base 10 logs below.

(x+2) .60206 = (2x-9) .77815

.60206x + 1.20412 = 1.55630x - 7.00336
-.95424x = - 8.20748
x = 8.60104

Solution:

log both sides and then solve.

so Log 4^(x+2)= Log 6^(2x-9) becomes (x+2) log 4 = (2x-9) log 6. Remember, log 4 and log 6 are just numbers, so let a = log 4 and b = log 6 then the above becomes

(x+2)*a = (2x-9)* b, now multiplying a and b through we get

ax + 2a = 2bx - 9b, rearranging terms and bringing x's to the same side and numbers to the other gives

ax-2bx = -9b - 2a this is equal to

x(a-2b) = -9b - 2a, therefore, x = (-9b - 2a) / (a -2b)

Ans a. 8.6

You need to convert to a common base. I'll use 4. Find a power of 4 that equals 6, as

4^b = 6 ... take log of both sides

log 4^b = log6 ... use the log rules

b(log 4) = log 6 ... divide by log 4

b = log6 / log 4 = 0.778 / 0.602 = 1.292 ... substitute back into original

4^(x+2) = (4^1.292)^(2x-9) ... exponents rule, multiply exponents

4^(x+2) = 4^(1.292(2x-9)) ... set the exponents equal and solve for x

x + 2 = 1.292(2x - 9) ... now it's simple algebra. solve this, you get x = 8.6

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