Find the solution of the exponential form in terms of logarithms

2^(2x+13) = 3^(x-45)

(2x+13) log 2 = (x -45) log 3

2x(log2 + 13log2 = x(log3) - 45log3
2x(log2) - x(log3) = -45log3 - 13log2
x(2log2 - log3) = -45log3 - 13log2
x = (-45log3 - 13log2)/(2log2 - log3)
x = -203.17

4e^x=91

2lnx-2+lny+.25lny

To find the solution of the exponential form in terms of logarithms, we can use the property that states that if two exponential expressions with the same base are equal, then their exponents are also equal. In this case, we have:

2^(2x + 13) = 3^(x - 45)

To solve for x, we can take the logarithm of both sides of the equation. The choice of the logarithm base is up to you, but the natural logarithm (ln) is commonly used. Applying the natural logarithm to both sides of the equation, we get:

ln(2^(2x + 13)) = ln(3^(x - 45))

Now, we can use the logarithm property that states that the exponent can be brought down as a multiplier:

(2x + 13) ln(2) = (x - 45) ln(3)

Expand the equation further:

2x ln(2) + 13 ln(2) = x ln(3) - 45 ln(3)

To isolate the x terms, we need to move all terms with x to one side and all terms without x to the other side:

2x ln(2) - x ln(3) = -13 ln(2) - 45 ln(3)

Now, factor out the x:

x (2 ln(2) - ln(3)) = -13 ln(2) - 45 ln(3)

Finally, divide both sides by (2 ln(2) - ln(3)):

x = (-13 ln(2) - 45 ln(3)) / (2 ln(2) - ln(3))

This is the solution of the exponential equation in terms of logarithms.