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To solve the equation \( 6 \cdot 4^{-t} = 5 \) for \( t \):

### Step 1: Isolate the Exponential Term
Divide both sides of the equation by \( 6 \):

\[
4^{-t} = \frac{5}{6}
\]

### Step 2: Take the Logarithm on Both Sides
To eliminate the exponent, take the natural logarithm (\( \ln \)) on both sides:

\[
\ln(4^{-t}) = \ln\left(\frac{5}{6}\right)
\]

### Step 3: Apply Logarithm Properties
Using the logarithmic rule \( \ln(a^b) = b \ln(a) \), the left-hand side becomes:

\[
-t \ln(4) = \ln\left(\frac{5}{6}\right)
\]

### Step 4: Solve for \( t \)
Now, divide both sides by \( -\ln(4) \):

\[
t = -\frac{\ln\left(\frac{5}{6}\right)}{\ln(4)}
\]

### Final Answer
Thus, the exact solution for \( t \) is:

\[
t = -\frac{\ln\left(\frac{5}{6}\right)}{\ln(4)}
\]

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

Here are some related questions to explore:
1. What are some common logarithmic properties used in solving exponential equations?
2. How would this equation change if the base 4 were replaced by another number, say 2?
3. How can we express the result using logarithms of a different base, such as base 2?
4. Why is it important to include the negative sign in the final answer for \( t \)?
5. Can we check if the answer is correct by plugging it back into the original equation?

**Tip:** When solving exponential equations, always consider isolating the exponential term first before applying logarithms!

Learn how to solve the exponential equation 6 * 4^(-t) = 5 for t by isolating the exponential term and applying logarithmic properties. This step-by-step solution covers algebraic concepts and is suitable for Grades 10-12.

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