Question

For each equation solve one of them by
using powers of the same number.to the
nearest hundredth, solve the other by
using logarithms. 8^x=3√2/4^x

Answer 1

Answer:

x = 0.42

Step-by-step explanation:

To solve the equation $8^x = \dfrac{3\sqrt{2}}{4^x}$ using logarithms, we can use the natural logarithm (ln) to maintain consistency.

Given:

$8^x = \dfrac{3\sqrt{2}}{4^x}$

Take the natural logarithm of both sides:

$\ln(8^x) = \ln\left(\dfrac{3\sqrt{2}}{4^x}\right)$

Using the property of logarithms that $\ln(a^b) = b \cdot \ln(a)$ and $\ln\left(\dfrac{a}{b}\right) = \ln(a) - \ln(b)$, we can simplify the equation:

$x \cdot \ln(8) = \ln(3\sqrt{2}) - x \cdot \ln(4)$

Now, isolate $x$ by moving all terms involving $x$ to one side of the equation:

$x \cdot \ln(8) + x \cdot \ln(4) = \ln(3\sqrt{2})$

Factor out $x$:

$x \cdot (\ln(8) + \ln(4)) = \ln(3\sqrt{2})$

Combine the logarithmic terms:

$x \cdot \ln(32) = \ln(3\sqrt{2})$

Now, solve for $x$:

$x = \dfrac{\ln(3\sqrt{2})}{\ln(32)}$

Using a calculator or computational tool:

$x \approx \dfrac{1.445185879 }{3.465735903}$

$x \approx 0.4169925001$

$x \approx 0.42 \textsf{(in nearest hundredth)}$

Therefore, the value of x is 0.42.