Math  /  Algebra

Question70(1.04)n=318570(1.04)^{n}=3185
Write your answers exactly. (i.e. no decimal approximations and simplified if possible), separated by a comma. n=n= \square Write your answers as decimals rounded to four decimal places. nn \approx \square

Studdy Solution

STEP 1

What is this asking? We're trying to find the number of times, nn, we need to multiply 70 by 1.04 to get 3185! Watch out! Remember, we can't just divide by 1.04 right away.
We need to isolate the exponential term first.
Also, don't forget to consider both exact and approximate solutions!

STEP 2

1. Isolate the exponential term
2. Apply the logarithm
3. Solve for nn
4. Calculate the approximate value

STEP 3

Alright, let's **kick things off** by getting (1.04)n(1.04)^n all by itself.
We've got 70(1.04)n=318570 \cdot (1.04)^n = 3185.
To **isolate** (1.04)n(1.04)^n, we need to **divide both sides** of the equation by **70**.
Remember, what we do to one side, we *must* do to the other to keep things balanced!

STEP 4

Dividing both sides by 70 gives us: 70(1.04)n70=318570 \frac{70 \cdot (1.04)^n}{70} = \frac{3185}{70} Simplifying, we get: (1.04)n=45.5 (1.04)^n = 45.5 Awesome! Now we have the exponential term isolated.

STEP 5

Now for the **magic of logarithms**!
We'll use the **natural logarithm (ln)** on both sides.
Why? Because logarithms help us bring down that exponent, nn, which is exactly what we want!

STEP 6

Applying the natural logarithm to both sides: ln((1.04)n)=ln(45.5) \ln((1.04)^n) = \ln(45.5) Using the **power rule of logarithms**, which says ln(ab)=bln(a)\ln(a^b) = b \cdot \ln(a), we can bring down the exponent: nln(1.04)=ln(45.5) n \cdot \ln(1.04) = \ln(45.5) We're getting closer!

STEP 7

Almost there!
Now we just need to **isolate** nn.
We can do this by **dividing both sides** by ln(1.04)\ln(1.04): nln(1.04)ln(1.04)=ln(45.5)ln(1.04) \frac{n \cdot \ln(1.04)}{\ln(1.04)} = \frac{\ln(45.5)}{\ln(1.04)} This simplifies to: n=ln(45.5)ln(1.04) n = \frac{\ln(45.5)}{\ln(1.04)} This is our **exact answer**!

STEP 8

Now, let's **crunch the numbers** to get a decimal approximation.
Using a calculator, we find: ln(45.5)3.81667945 \ln(45.5) \approx 3.81667945 and ln(1.04)0.03922071 \ln(1.04) \approx 0.03922071 Therefore, n3.816679450.0392207197.31275 n \approx \frac{3.81667945}{0.03922071} \approx 97.31275 Rounding to four decimal places, we get **97.3128**.

STEP 9

Exact answer: ln(45.5)ln(1.04)\frac{\ln(45.5)}{\ln(1.04)} Approximate answer: 97.3128

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