Question

solve for x to the nearest hundreth
56times2^2x=22

Answer 1

To solve the equation \( 56 \times 2^{2x} = 22 \), we can start by isolating \( 2^{2x} \).

1. Divide both sides by 56:

\[
2^{2x} = \frac{22}{56}
\]

2. Simplify the fraction:

\[
\frac{22}{56} = \frac{11}{28}
\]

So we have:

\[
2^{2x} = \frac{11}{28}
\]

3. Now, we apply the logarithm to both sides. We can use any logarithm, but we'll use the natural logarithm \( \ln \):

\[
\ln(2^{2x}) = \ln\left(\frac{11}{28}\right)
\]

4. By using the power rule of logarithms:

\[
2x \cdot \ln(2) = \ln(11) - \ln(28)
\]

5. Solve for \( 2x \):

\[
2x = \frac{\ln(11) - \ln(28)}{\ln(2)}
\]

6. Now, divide both sides by 2 to find \( x \):

\[
x = \frac{\ln(11) - \ln(28)}{2\ln(2)}
\]

7. Now, plug in the values of the logarithms (approximately):

- \( \ln(11) \approx 2.3979 \)
- \( \ln(28) \approx 3.3322 \)
- \( \ln(2) \approx 0.6931 \)

Calculating \( \ln(11) - \ln(28) \):

\[
\ln(11) - \ln(28) \approx 2.3979 - 3.3322 \approx -0.9343
\]

Now, substituting back into the equation for \( x \):

\[
x \approx \frac{-0.9343}{2 \times 0.6931} \approx \frac{-0.9343}{1.3862} \approx -0.6735
\]

8. Finally, rounding to the nearest hundredth:

\[
\boxed{-0.67}
\]