(3^x)(4^2x+1)=(6^x+2). Find the exact value of x giving the answer in the form x=ln(a)/ln(b).

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

Thank you so much! I got to the third step and then couldn't figure out how to go about it without using decimals..but thanks so much!

To solve this equation, we can take the logarithm of both sides of the equation. Applying logarithms helps us simplify the equation by bringing down the exponent:

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).