solve each equation. round your answers to the nearest ten-thousandth -9*12^5k = -87

We are given the equation:

\[
-9 \times 12^{5k} = -87
\]

To solve for \( k \), follow these steps:

### Step 1: Divide both sides by -9 to isolate the exponential term.

\[
12^{5k} = \frac{-87}{-9}
\]

This simplifies to:

\[
12^{5k} = 9.6667
\]

### Step 2: Take the natural logarithm (ln) of both sides to deal with the exponent.

\[
\ln(12^{5k}) = \ln(9.6667)
\]

Using the logarithmic identity \(\ln(a^b) = b \ln(a)\), this becomes:

\[
5k \ln(12) = \ln(9.6667)
\]

### Step 3: Solve for \( k \).

\[
k = \frac{\ln(9.6667)}{5 \ln(12)}
\]

Let's compute the value of \( k \). 

The value of \( k \), rounded to the nearest ten-thousandth, is \( k \approx 0.1826 \).

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How do logarithms help in solving exponential equations?
2. What are the steps to take when isolating variables in an equation?
3. How does rounding affect the precision of your solution?
4. What are the properties of logarithms that can simplify solving exponential equations?
5. How can you apply these steps to solve more complex exponential equations?

**Tip:** Always check if dividing both sides of an equation by a negative number changes the signs appropriately.

Learn how to solve the exponential equation -9*12^5k = -87 step-by-step using logarithms. This problem includes key algebraic concepts, such as exponentiation, logarithmic identities, and rounding answers to the nearest ten-thousandth. Suitable for 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