2^(2x+14)=3^(x-17)

Question

Could someone please explain to me the steps to solving this problem. I know it has something to do with logarithms.

Answer 1

Another approach is to take the natural log of both sides:

 

ln(2^2x+14) = ln(3^x-17)

 

So, (2x+14)ln2 = (x-17)ln3

 

2xln2 + 14ln2 = xln3 - 17ln3

 

2xln2-xln3 = -14ln2-17ln3

 

x(2ln2-ln3) = -14ln2-17ln3

 

x = (-14ln2-17ln3)/(2ln2-ln3)

 

x≈-98.65

 

 

Answer 2

In this example we have the problem:

 

2^(2x+14) = 3^(x-17)

 

In this case, we pick one of the bases. For this example, I'll pick the lefthand base of 2

 

log₂(2^(2x+14)) = log₂(3^(x-17))

 

The left hand logarithm and base cancel, giving us 2x + 14, the righthand item needs to use the exponent property of logarithms, which states that log_b(xⁿ) = n * log_b(x)

 

2x + 14 = (x - 17) * log₂(3)

 

I now distribute and move the x-termed items to the left and the constant terms to the right.

 

(2x - log₂(3) * x) = (-17 * log₂(3) - 14)

 

I factor out the x on the left hand side to get:

 

x(2 - log₂(3)) = (-17 * log₂(3) - 14)

 

and divide the non-x term:

 

x = (-17 * log₂(3) - 14) / (2 - log₂(3))

 

Calculating gives us:

 

x = -98.65

 

If google is to be believed with their calculator.