To solve the inequality \( 150 > 10^x - 2 \), follow these steps:

### Step 1: Isolate \( 10^x \)
Add 2 to both sides of the inequality:

\[
150 + 2 > 10^x
\]
\[
152 > 10^x
\]

### Step 2: Take the logarithm of both sides
We take the logarithm (base 10) of both sides to deal with the exponent:

\[
\log_{10}(152) > \log_{10}(10^x)
\]

### Step 3: Apply the logarithmic identity
Using the identity \( \log_{10}(10^x) = x \), we simplify the right-hand side:

\[
\log_{10}(152) > x
\]

### Step 4: Calculate \( \log_{10}(152) \)
Now calculate \( \log_{10}(152) \):

\[
\log_{10}(152) \approx 2.1818
\]

### Final Answer:
So, the solution is:

\[
x < 2.1818
\]

This means \( x \) must be less than approximately 2.18 for the inequality to hold true.

Would you like more details or have any questions about this?

### Related Questions:
1. How would the solution change if the inequality were \( 150 \geq 10^x - 2 \)?
2. What if the inequality involved a natural logarithm instead of a base 10 logarithm?
3. How does the base of the logarithm affect the solution of inequalities like this one?
4. How can we express this inequality in terms of exponential functions?
5. Can inequalities like this be solved graphically, and if so, how?

**Tip:** When solving inequalities involving exponents, isolating the exponential term is key before applying logarithms.

This step-by-step solution walks through solving the inequality 150 > 10^x - 2 using logarithms. Learn how to isolate the exponential term and apply the logarithmic identity to find that x must be less than approximately 2.18. Suitable for Grades 9-11 students.

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