Question

solve the equation. round answers to four decimal places.
$1.57^{4x}=2$

$x\approx\square$
(round to four decimal places as needed.)

Explanation:

Step1: Take natural log on both sides

Take the natural logarithm of both sides of the equation \(1.57^{4x}=2\). Using the property of logarithms \(\ln(a^b)=b\ln(a)\), we get:
$$\ln(1.57^{4x})=\ln(2)$$
$$4x\ln(1.57)=\ln(2)$$

Step2: Solve for x

Divide both sides by \(4\ln(1.57)\) to isolate \(x\):
$$x = \frac{\ln(2)}{4\ln(1.57)}$$
Now, calculate the values. We know that \(\ln(2)\approx0.6931\) and \(\ln(1.57)\approx0.4510\).
Substitute these values into the formula:
$$x=\frac{0.6931}{4\times0.4510}=\frac{0.6931}{1.804}\approx0.3842$$

Answer:

\(x\approx0.3842\)