Exponent Equation, Quick Answer Please?

log 5^(x+2) = log 3^(2x+1) <- Take the log of both sides

(x+2) log 5 = (2x+1) log 3 <- Bring the exponents down as the coefficient

x log 5 + 2 log 5 = 2x log 3 + log 3 <- Distribute

x log 5 - 2x log 3 = log 3 - 2 log 5 <- Bring all terms with 'x' to one side

x (log 5 - 2 log 3) = log 3 - 2 log 5 <- Factor out x

x = (log 3 - 2 log 5) / (log 5 - 2 log 3) <- Divide

x = about 3.61

5^(x+2) = 3^(2x+1)

Pull out the constant part of each exponent...

5^2 * 5^x = 3 * 3^2x

Normalize exponents to be x:

25 * 5^x = 3 * 9^x

Cross-multiply:

25/3 = (9/5)^x

Take log base 9/5 of both sides:

log(9/5) 25/3 = x

Because log(a) b = log(10) b / log(10) a...

log(10) (25/3) / log(10) (9/5) = x

0.9208 / 0.2553 = x

x = 3.607

Take the log of both sides:

log(5^(x+2)) = log(3^(2x+1))

log(a^b) = b * log(a)

(x + 2) * log(5) = (2x + 1) * log(3)

Solve normally:

(x + 2) = log(3)/log(5) * (2x + 1)

x + 2 = log(3)/log(5) * 2x + log(3)/log(5)

(1 - 2log(3)/log(5)) * x = log(3)/log(5) - 2

x = (log(3)/log(5) - 2)/(1 - 2log(3)/log(5))

x = 3.61(approximately)

5^(x + 2) = 3^(2x + 1)

(x + 2) Log 5 = (2x + 1) Log 3

0.699(x + 2) = 0.477(2x + 1)

0.699x + 1.4 = 0.954x + 0.477

0.699x - 0.954x = 0.477 - 1.4

- 0.255x = - 0.923

x = - 0.923 / - 0.255

x = 3.62

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