To solve the given equation [tex]\( 3^n = 1519 \)[/tex] for [tex]\( n \)[/tex], we need to use logarithms and the change of base formula. Here are the steps in detail:
1. Convert the equation to logarithmic form:
The equation [tex]\( 3^n = 1519 \)[/tex] can be written in logarithmic form using the properties of logarithms. Specifically, we take the logarithm of both sides of the equation. Typically, we use the natural logarithm ([tex]\(\log\)[/tex]) or the common logarithm ([tex]\(\log_{10}\)[/tex]) for convenience. Here, we will take the natural logarithm ([tex]\(\log\)[/tex]) of both sides:
[tex]\[
\log(3^n) = \log(1519)
\][/tex]
2. Use the power rule of logarithms:
The power rule states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]. Applying this to the left side of the equation, we get:
[tex]\[
n \cdot \log(3) = \log(1519)
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
Now, isolate [tex]\( n \)[/tex] by dividing both sides of the equation by [tex]\(\log(3)\)[/tex]:
[tex]\[
n = \frac{\log(1519)}{\log(3)}
\][/tex]
4. Apply the change of base formula:
According to the change of base formula, [tex]\(\log_b(a) = \frac{\log(a)}{\log(b)}\)[/tex], where [tex]\( a = 1519 \)[/tex] and [tex]\( b = 3 \)[/tex]. Plug in the values into the formula to find [tex]\( n \)[/tex]:
[tex]\[
n = \frac{\log(1519)}{\log(3)}
\][/tex]
5. Calculate the value of [tex]\( n \)[/tex]:
Using a calculator, we find:
[tex]\[
\log(1519) \approx 3.181771
\][/tex]
[tex]\[
\log(3) \approx 0.477121
\][/tex]
So:
[tex]\[
n \approx \frac{3.181771}{0.477121} \approx 6.668
\][/tex]
6. Round the answer to 3 decimal places:
The value of [tex]\( n \)[/tex], accurate to three decimal places, is:
[tex]\[
n \approx 6.668
\][/tex]
Thus, the value of [tex]\( n \)[/tex] is [tex]\( 6.668 \)[/tex] when rounded to three decimal places.
