solve the exponential equation.
4^(x+2)=9x
A rough sketch of
y = 4^(x+2) and y = 9x shows that the two can never meet.
So, there is no solution
Did you mean:
4^(x+2) = 9^x ?
yes i meant 4^(x+2)=9^x
ok, then take log of both sides
log (4^(x+2) ) = log (9^x)
(x+2)log4 = xlog9
xlog4 + 2log4= xlog9
2log4 = xlog9 - xlog4
x(log9-log4) = 2log4
x = 2log4/(log9-log4)
button-pushing time ....
x = appr 3.419
To solve the exponential equation 4^(x+2) = 9x, we need to isolate the variable x.
Step 1: Simplify the equation.
4^(x+2) can be rewritten as (2^2)^(x+2), which simplifies to 2^(2x+4). Rewriting 9x as (3^2)x gives us 3^(2x).
So, the equation becomes 2^(2x+4) = 3^(2x).
Step 2: Take the logarithm of both sides.
To simplify the equation further, take the logarithm of both sides. The choice of logarithm base is up to you, but usually, it is convenient to use a common logarithm (log base 10) or a natural logarithm (log base e).
We'll use the natural logarithm, denoted as "ln."
Taking the natural logarithm of both sides, we have ln(2^(2x+4)) = ln(3^(2x)).
Using the logarithmic property, we can bring down the exponents:
(2x + 4)ln(2) = (2x)ln(3).
Step 3: Solve for x.
Now, we can solve for x. Distribute the ln(2) on the left side of the equation and the ln(3) on the right side:
2xln(2) + 4ln(2) = 2xln(3).
Rearrange the equation to isolate the x terms on one side:
2xln(2) - 2xln(3) = -4ln(2).
Factor out x:
x(2ln(2) - 2ln(3)) = -4ln(2).
Divide both sides by (2ln(2) - 2ln(3)):
x = -4ln(2) / (2ln(2) - 2ln(3)).
Simplifying further, we get:
x = -2ln(2) / (ln(2) - ln(3)).
This is the solution to the exponential equation 4^(x+2) = 9x.