To solve the inequality $5^x + 3 \geq 25$, follow these steps:

Step 1: Simplify the inequality

Subtract 3 from both sides: $5^x \geq 22$

Step 2: Take the logarithm (logarithmic form)

Use the property of logarithms: $a^b = c \implies b = \log_a(c)$.

Take the base-5 logarithm of both sides: $x \geq \log_5(22)$

Step 3: Convert logarithms to base 10 (optional)

Using the change-of-base formula $\log_a(b) = \frac{\log(b)}{\log(a)}$, we rewrite: $x \geq \frac{\log(22)}{\log(5)}$

Step 4: Approximate the value

Using a calculator: $\log(22) \approx 1.3424, \quad \log(5) \approx 0.69897$ $x \geq \frac{1.3424}{0.69897} \approx 1.92$

Final Answer:

$x \geq 1.92$

Let me know if you'd like a detailed breakdown of the logarithmic steps or have questions!

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Related Questions:

1. How would the solution change if the inequality were $5^x + 3 \leq 25$?
2. What is the general approach to solving $a^x + b \geq c$?
3. How do you graph $5^x + 3$ and $25$ to verify the solution visually?
4. Can this inequality be solved using a different base for logarithms, like natural logarithms?
5. What is the role of logarithms in solving exponential equations and inequalities?

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Tip:

When solving exponential equations or inequalities, always isolate the exponential term first for easier application of logarithmic rules.