Solve the exponential equation using logarithms. Give the answer in decimal form, rounding to four decimal places. (Enter your answers as a comma-separated list.)
4^(x − 3) = 3^(2x)
x = _____
4^(x-3) = 3^(2x)
(x-3)*Log4 = 2x*Log 3
Divide both sides by Log 4:
x-3 = 2x * 0.79248 = 1.5850x
x - 1.5850x = 3
-0.5850x = 3
X = -5.1282
To solve the equation 4^(x-3) = 3^(2x) using logarithms, we can take the logarithm of both sides of the equation.
Step 1: Take the logarithm of both sides. We can use either the natural logarithm (ln) or the common logarithm (log).
ln(4^(x-3)) = ln(3^(2x))
Step 2: Apply the properties of logarithms to simplify the equation.
(x-3) * ln(4) = 2x * ln(3)
Step 3: Distribute the ln(4) and ln(3) to the terms inside the parentheses.
x * ln(4) - 3 * ln(4) = 2x * ln(3)
Step 4: Move all the terms involving x to one side of the equation.
x * ln(4) - 2x * ln(3) = 3 * ln(4)
Step 5: Factor out x from the left side of the equation.
x (ln(4) - 2 ln(3)) = 3 * ln(4)
Step 6: Divide both sides of the equation by (ln(4) - 2 ln(3)).
x = (3 * ln(4)) / (ln(4) - 2 ln(3))
Step 7: Use a calculator to evaluate the expression on the right side of the equation. Round the answer to four decimal places.
x ≈ 2.1230
Therefore, the solution to the equation 4^(x-3) = 3^(2x) is x ≈ 2.1230.