solve for $x$, rounding to the nearest hundredth.\n$12 \\cdot 10^{\\frac{x}{4}} = 43$

Explanation:

Step1: Isolate the exponential term

Divide both sides by 12. $$10^{\frac{x}{4}} = \frac{43}{12}$$

Step2: Convert to logarithmic form

Take base-10 log of both sides. $$\frac{x}{4} = \log_{10}\left(\frac{43}{12}\right)$$

Step3: Solve for x

Multiply both sides by 4. $$x = 4 \cdot \log_{10}\left(\frac{43}{12}\right)$$

Step4: Calculate the value

Compute the log and multiply, then round. $$x = 4 \cdot \log_{10}(3.5833) \approx 4 \cdot 0.5543 = 2.2172$$

Answer:

$2.22$