The equation given is:

\[
160,000 = 80,000(1.09)^t
\]

We need to solve for \( t \).

### Step 1: Simplify the equation
Divide both sides of the equation by 80,000:

\[
\frac{160,000}{80,000} = (1.09)^t
\]

This simplifies to:

\[
2 = (1.09)^t
\]

### Step 2: Solve for \( t \) using logarithms
Take the natural logarithm (ln) of both sides:

\[
\ln(2) = \ln((1.09)^t)
\]

Using the logarithmic property \(\ln(a^b) = b\ln(a)\), we get:

\[
\ln(2) = t \cdot \ln(1.09)
\]

### Step 3: Isolate \( t \)

Solve for \( t \) by dividing both sides by \(\ln(1.09)\):

\[
t = \frac{\ln(2)}{\ln(1.09)}
\]

### Step 4: Calculate the value of \( t \)

Now we can compute the value:

\[
t \approx \frac{0.6931}{0.0862} \approx 8.04
\]

So, \( t \approx 8.04 \).

### Final Answer:
The value of \( t \) is approximately **8.04 years**.

Would you like more details on logarithmic properties or have any other questions?

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Here are 5 related questions:

1. How do we calculate natural logarithms manually?
2. Why do we use logarithms to solve exponential equations?
3. What is the meaning of the base in an exponential function?
4. Can you solve a similar equation with a different base, like 1.05?
5. What happens if we change the initial values in this type of equation?

**Tip**: When dealing with exponential growth or decay, logarithms are the key to "undo" the exponential and solve for the variable.

Learn how to solve the exponential equation 160000 = 80000(1.09)^t using logarithms. This problem demonstrates key mathematical concepts such as exponential growth and the logarithmic property. Ideal for students in Grades 9-12.

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