1. **State the problem:** Find the value of $x$ that satisfies the equation $$10 = 23^{x+1}$$. 2. **Rewrite the equation:** The equation is already in the form $$10 = 23^{x+1}$$. 3. **Isolate the exponential term:** To solve for $x$, we want to isolate $x$ in the exponent. First, take the logarithm of both sides. We can use the natural logarithm (ln) or log base 10. Here, we use natural logarithm: $$\ln(10) = \ln\left(23^{x+1}\right)$$ 4. **Use logarithm power rule:** The logarithm of a power is the exponent times the logarithm of the base: $$\ln(10) = (x+1) \ln(23)$$ 5. **Solve for $x$:** Divide both sides by $\ln(23)$: $$\frac{\ln(10)}{\ln(23)} = x + 1$$ 6. **Isolate $x$:** $$x = \frac{\ln(10)}{\ln(23)} - 1$$ 7. **Calculate the numerical value:** Using approximate values: $$\ln(10) \approx 2.302585$$ $$\ln(23) \approx 3.135494$$ So, $$x \approx \frac{2.302585}{3.135494} - 1 \approx 0.734 - 1 = -0.266$$ **Final answer:** $$x \approx -0.266$$