Use your log rules
(3^x)(4^2x+1)=(6^x+2)
log[(3^x)(4^2x+1)]= log(6^x+2)
xlog3 + (2x+1)log4 = (x+2)log6
xlog3 + x(2log4) + log4 = xlog6 + 2log4
x(log3 + 2log4 - log6) = 2log4 - log4
x(log8) = log9
x = log9/log8
(3^x)(4^2x+1)=(6^x+2)
log[(3^x)(4^2x+1)]= log(6^x+2)
xlog3 + (2x+1)log4 = (x+2)log6
xlog3 + x(2log4) + log4 = xlog6 + 2log4
x(log3 + 2log4 - log6) = 2log4 - log4
x(log8) = log9
x = log9/log8
ln[(3^x)(4^(2x+1))] = ln(6^(x+2))
Now, we can apply logarithmic properties to further simplify the equation. In particular, we can use the power rule of logarithms and rewrite the equation as:
ln(3^x) + ln(4^(2x+1)) = ln(6^(x+2))
Applying the power rule, we can bring down the exponents as coefficients:
x * ln(3) + (2x + 1) * ln(4) = (x + 2) * ln(6)
Next, we can distribute the logarithms:
x * ln(3) + 2x * ln(4) + ln(4) = x * ln(6) + 2 * ln(6)
Now, let's collect like terms:
x * [ln(3) - ln(6)] + 2x * ln(4) - 2 * ln(6) = -ln(4)
Now, we can combine the coefficients of x and apply logarithmic properties:
x * [ln(3/6)] + ln(4^(2x)) - ln(6^2) = -ln(4)
Simplifying inside the logarithms:
x * [ln(1/2)] + ln(2^(4x)) - ln(36) = -ln(4)
Next, recall that ln(1/2) = -ln(2). Applying this:
- x * ln(2) + ln(2^(4x)) - ln(36) = -ln(4)
The ln(2) terms cancel out, giving us:
ln(2^(4x))/36 = ln(4)
Now, if the logarithms of two numbers are equal, the numbers themselves must be equal:
2^(4x) = 4^36
Simplifying further:
2^(4x) = 2^(2 * 36)
Since the bases are the same, we can equate the exponents:
4x = 2 * 36
4x = 72
Dividing both sides by 4:
x = 72/4
x = 18
Therefore, the exact value of x is x = ln(18) / ln(2).