SOLUTION: Find the exact solution, using common logarithms. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 4^(2x + 3) = 5^(x

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Question 1005203: Find the exact solution, using common logarithms. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
4^(2x + 3) = 5^(x - 2)
Find a two-decimal-place approximation of each solution. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
start with 4^(2x+3) = 5^(x - 2)

take the log of both sides of the equation to get:

log(4^(2x+3)) = log(5^(x-2))

since log(a^b) = b*log(a), apply this property to your expression to get:

(2x+3) * log(4) = (x-2) * log(5)

use the distributive law of multiplication to get:

2x * log(4) + 3 * log(4) = x * log(5) - 2 * log(5)

subtract x * log(5) from both sides of the equation and subtract 3 * log(4) from both sides of the equation to get:

2x * log(4) - x * log(5) = - 2 * log(5) - 3 * log(4)

factor out the x on the left side of the equation to get:

x * (2 * log(4) - log(5)) = -2 * log(5) - 3 * log(4)

divide both sides of the equation by (2 * log(4) - log(5)) to get:

x = (-2 * log(5) - 3 * log(4)) / (2 * log(4) - log(5))

use your calculator to evaluate the expression on the right side of the equation.

you will get x = -6.342908285

replace x in your original equation with that value and you will get:

4^(2x+3) = 5^(x - 2) becomes 4^(2 * -6.342908285 + 3) = 5^(-6.342908285 - 2)

perform the evaluation and you will get:

1.474203235 * 10^-6 = 1.474203235 * 10^-6.

since they're equal, the value for x is good.

Answer by MathTherapy(10551) (Show Source):
You can put this solution on YOUR website!

Find the exact solution, using common logarithms. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
4^(2x + 3) = 5^(x - 2)
Find a two-decimal-place approximation of each solution. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

-------- Taking the log of both sides
-------- Applying

2x + 3 = 1.160964047(x - 2) -------- Cross-multiplying
2x + 3 = 1.160964047x - 2.321928
2x - 1.160964047x = - 2.321928 - 3
.839035953x = - 5.321928
, or

SOLUTION: Find the exact solution, using common logarithms. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 4^(2x + 3) = 5^(x - 2) Find a tw