To solve the equation:

\[
6^{(x-2)} = 5^{(2x+3)}
\]

we can use logarithms to bring down the exponents. Here's the step-by-step solution:

### Step 1: Take the logarithm of both sides.
We can take the logarithm with any base, but the natural logarithm (ln) is commonly used:

\[
\ln\left(6^{(x-2)}\right) = \ln\left(5^{(2x+3)}\right)
\]

### Step 2: Use the power rule of logarithms.
The power rule states that \(\ln(a^b) = b \cdot \ln(a)\), so we can move the exponents in front:

\[
(x-2) \cdot \ln(6) = (2x+3) \cdot \ln(5)
\]

### Step 3: Distribute the logarithms.
Expand both sides of the equation:

\[
x \cdot \ln(6) - 2 \cdot \ln(6) = 2x \cdot \ln(5) + 3 \cdot \ln(5)
\]

### Step 4: Collect like terms.
Group the terms involving \(x\) on one side and the constant terms on the other side:

\[
x \cdot \ln(6) - 2x \cdot \ln(5) = 3 \cdot \ln(5) + 2 \cdot \ln(6)
\]

Factor out \(x\) from the left side:

\[
x \left(\ln(6) - 2 \cdot \ln(5)\right) = 3 \cdot \ln(5) + 2 \cdot \ln(6)
\]

### Step 5: Solve for \(x\).
Finally, solve for \(x\) by dividing both sides by \(\ln(6) - 2 \cdot \ln(5)\):

\[
x = \frac{3 \cdot \ln(5) + 2 \cdot \ln(6)}{\ln(6) - 2 \cdot \ln(5)}
\]

### Step 6: Approximate the solution (optional).
To get a numerical approximation, you can use the values of \(\ln(5)\) and \(\ln(6)\):

\[
\ln(5) \approx 1.60944, \quad \ln(6) \approx 1.79176
\]

Substitute these values:

\[
x \approx \frac{3 \times 1.60944 + 2 \times 1.79176}{1.79176 - 2 \times 1.60944}
\]

Calculate the expression:

\[
x \approx \frac{4.82832 + 3.58352}{1.79176 - 3.21888} \approx \frac{8.41184}{-1.42712} \approx -5.896
\]

So the approximate solution is:

\[
\boxed{x \approx -5.896}
\]

Would you like a more detailed breakdown or have any questions?

---

Here are five related questions:
1. How do logarithmic identities help simplify complex equations?
2. Can you solve \(4^{(x+1)} = 3^{(3x-2)}\)?
3. What methods can be used to solve exponential equations without logarithms?
4. How do you solve equations where logarithms and exponents appear together?
5. What happens if the bases in the exponentials are the same?

**Tip:** When solving exponential equations, logarithms are particularly useful because they allow you to bring down the exponents, making the equation easier to solve.

Learn how to solve the exponential equation 6^(x-2) = 5^(2x+3) with a step-by-step solution involving logarithms. This problem covers exponential equations and logarithmic methods, suitable for advanced high school students and those preparing for mathematical competitions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation