https://file.aiquickdraw.com/mathbot/file/1729181887819u6joy05t.jpg

The equation from your image is:

\[
\frac{1}{5}^x - 10 = 4
\]

Let's solve this step by step.

### Step 1: Isolate the exponential term.
Add 10 to both sides:

\[
\frac{1}{5}^x = 14
\]

### Step 2: Rewrite the base.
\(\frac{1}{5} = 5^{-1}\), so:

\[
\left(5^{-1}\right)^x = 14
\]

Which simplifies to:

\[
5^{-x} = 14
\]

### Step 3: Take the logarithm of both sides.
We can use the natural logarithm (ln) or common logarithm (log). I'll use natural logarithm:

\[
\ln\left(5^{-x}\right) = \ln(14)
\]

### Step 4: Apply the logarithm rules.
Using the rule \(\ln(a^b) = b \cdot \ln(a)\), we get:

\[
-x \cdot \ln(5) = \ln(14)
\]

### Step 5: Solve for \(x\).
Divide both sides by \(-\ln(5)\):

\[
x = -\frac{\ln(14)}{\ln(5)}
\]

### Step 6: Calculate the value.
Using a calculator:

\[
\ln(14) \approx 2.639
\quad \text{and} \quad \ln(5) \approx 1.609
\]

\[
x = -\frac{2.639}{1.609} \approx -1.64
\]

Thus, the solution is:

\[
x \approx -1.64
\]

---

Would you like further details or clarifications?

### 5 Related Questions:
1. What is the difference between natural logarithm and common logarithm?
2. How do logarithmic rules apply to exponential equations?
3. Can you solve this equation graphically instead of algebraically?
4. What are other ways to solve equations involving negative exponents?
5. How does the base of the exponential function affect the solution?

### Tip:
When working with equations involving negative exponents, rewriting terms (like \( \frac{1}{5} = 5^{-1} \)) can make the problem easier to manage.

Learn how to solve the exponential equation (1/5)^x - 10 = 4 step by step. This detailed guide covers the application of logarithms, exponential rules, and negative exponents. Suitable for high school students in Grades 10-12.

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