Question 385067
{{{4^(2x+1)=8^(x+4)}}}
With variables in exponents we would usually use logarithms to solve an equation like this. And we could do so on this equation. However, since 4 and 8 are both powers of 2 we can express both sides of the equation as powers of 2. Such an equation is easier to solve this way than with logarithms:
{{{(2^2)^(2x+1) = (2^3)^(x+4)}}}
When raising a power to a power, the rule for the exponents is to multiply them. So the equation simplifies as follows:
{{{2^(2*(2x+1)) = 2^(3*(x+4))}}}
{{{2^(4x+2) = 2^(3x+12)}}}
Now the equation says that two powers of 2 are equal. The only way this can be true is if the exponents are equal, too. So:
4x + 2 = 3x + 12
This is easy to solve. Subtract 3x from each side:
x + 2 = 12
Subtract 2 from each side:
x = 10