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------------------------------------------------ Problem: Solve [tex]$20000=1500 e^{0.04 t}$[/tex] for [tex]$t$[/tex].

Sure! Let's solve the equation [tex]$20000 = 1500 e^{0.04t}$[/tex] for [tex]$t$[/tex].

### Step-by-Step Solution:

1. Understand the Equation:

The equation given is [tex]$20000 = 1500 e^{0.04t}$[/tex], which is in the form of [tex]$A = Pe^{rt}$[/tex], where:
- [tex]$A = 20000$[/tex] is the final amount.
- [tex]$P = 1500$[/tex] is the initial amount.
- [tex]$r = 0.04$[/tex] is the rate.
- [tex]$t$[/tex] is the time we are solving for.

2. Isolate the Exponential Term:

Divide both sides of the equation by 1500 to isolate the exponential term:

[tex]\[
\frac{20000}{1500} = e^{0.04t}
\][/tex]

Simplify the left side:

[tex]\[
\frac{20000}{1500} = \frac{2000}{150} = \frac{200}{15} \approx 13.3333
\][/tex]

So the equation becomes:

[tex]\[
e^{0.04t} = 13.3333
\][/tex]

3. Solve for [tex]$t$[/tex] using Natural Logarithms:

To solve for [tex]$t$[/tex], take the natural logarithm (ln) on both sides:

[tex]\[
\ln(e^{0.04t}) = \ln(13.3333)
\][/tex]

By the properties of logarithms, the left side simplifies to:

[tex]\[
0.04t = \ln(13.3333)
\][/tex]

4. Solve for [tex]$t$[/tex]:

Divide both sides by 0.04 to isolate [tex]$t$[/tex]:

[tex]\[
t = \frac{\ln(13.3333)}{0.04}
\][/tex]

5. Calculate [tex]$t$[/tex]:

Using a calculator, compute [tex]$ \ln(13.3333) \approx 2.5903 $[/tex].

Now, divide by 0.04:

[tex]\[
t = \frac{2.5903}{0.04} \approx 64.7567
\][/tex]

The solution is:

[tex]\[ t \approx 64.76 \][/tex]

Therefore, it takes approximately 64.76 years for the initial amount to grow to $20,000 at a rate of 0.04.