MHB How Many Times Do I Have To Increase by 3%?

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To determine how many times to increase 30 by 3% to reach or exceed 500, the formula used is 30 * 1.03^n ≥ 500. By dividing through by 30 and taking the natural logarithm, the equation simplifies to n ≥ ln(50/3) / ln(1.03). The result indicates that n must be rounded up to the nearest integer, yielding n = 96. This means 30 must be increased by 3% a total of 96 times to meet or exceed 500.
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I want to know how many time I have to increase 30 by 3% before it is greater or equal to 500.

I think this is the correct formula:
30 * 1.03^x >= 500

What steps do I have to take to solve it?
 
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I would write:

$$30\cdot1.03^n\ge500$$

Divide through by 30:

$$1.03^n\ge\frac{50}{3}$$

Take the natural log of both sides, and apply a log rule to obtain:

$$n\ln(1.03)\ge\ln\left(\frac{50}{3}\right)$$

Hence:

$$n\ge\frac{\ln\left(\dfrac{50}{3}\right)}{\ln(1.03)}$$

Since presumably \(n\) is an integer, we could write:

$$n=\left\lceil\frac{\ln\left(\dfrac{50}{3}\right)}{\ln(1.03)}\right\rceil$$

This is the "ceiling" function and it tells us to round up to the nearest integer. According to W|A, we find:

$$n=96$$

Wolfram|Alpha: Computational Intelligence
 
piAreRound said:
I want to know how many time I have to increase 30 by 3% before it is greater or equal to 500.
Tattoo this on your wrist (under your watch!):

if a^p = b then p = ln(b) / ln(a)
 
My wrist is getting awfully crowded!
 
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