Question23. Solve the equation. If necessary, round the answer to four decimal places. $$7^{x+2}-7^{x}=96$$

Studdy Solution
To solve for $x$, take the logarithm of both sides. We will use the natural logarithm (ln) for this step:
$$\ln(7^x) = \ln(2)$$
Apply the power rule of logarithms:
$$x \cdot \ln(7) = \ln(2)$$
Solve for $x$ by dividing both sides by $\ln(7)$:
$$x = \frac{\ln(2)}{\ln(7)}$$
Calculate the value of $x$ and round it to four decimal places:
$$x \approx \frac{0.6931}{1.9459}$$ $$x \approx 0.3562$$
The value of $x$, rounded to four decimal places, is:
$$\boxed{0.3562}$$