https://file.aiquickdraw.com/mathbot/file/17300846527705us3532m.jpg

To solve the equation \( 6^{x - 5} = 38 \) for \( x \), follow these steps:

### Step 1: Take the natural logarithm of both sides
Taking the natural logarithm (ln) on both sides will help bring down the exponent:
\[
\ln(6^{x - 5}) = \ln(38)
\]

### Step 2: Apply the power rule for logarithms
Using the rule \( \ln(a^b) = b \ln(a) \), we can rewrite the left side as:
\[
(x - 5) \ln(6) = \ln(38)
\]

### Step 3: Solve for \( x \)
Now, divide both sides by \( \ln(6) \) to isolate \( x - 5 \):
\[
x - 5 = \frac{\ln(38)}{\ln(6)}
\]

Then, add 5 to both sides to solve for \( x \):
\[
x = \frac{\ln(38)}{\ln(6)} + 5
\]

### Step 4: Calculate the result
Using approximate values:
\[
\ln(38) \approx 3.6376 \quad \text{and} \quad \ln(6) \approx 1.7918
\]
\[
x \approx \frac{3.6376}{1.7918} + 5
\]
\[
x \approx 2.0306 + 5
\]
\[
x \approx 7.0306
\]

### Final Answer
\[
x \approx 7.0306
\]

Would you like further details or have any questions about this solution?

### Related Questions:
1. How would you solve \( 6^{2x - 4} = 45 \)?
2. What happens if the base in the equation was 10 instead of 6?
3. Can you solve the equation \( 7^{x - 3} = 20 \) in a similar manner?
4. What if the equation was \( 6^{x - 5} = e^{2x} \)?
5. How does taking the natural logarithm help in solving exponential equations?

### Tip:
When solving exponential equations, taking the logarithm (usually natural log) of both sides is a key strategy to bring down the exponent and simplify the equation.

Solve the given equation for x: 6^(x-5) = 38

Learn how to solve the exponential equation 6^(x-5) = 38 for x using logarithmic properties. This problem covers exponential and logarithmic concepts, including applying the power rule for logarithms. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation