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Chapter 4: Problem 27

Find a number \(r\) such that $$ \left(1+\frac{r}{10^{90}}\right)^{\left(10^{90}\right)} \approx 5 $$

Short Answer

Expert verified
The value of \(r\) that makes the given expression close to 5 is approximately 1.609.

Step by step solution

01

Rewrite the given expression in terms of r

Lets rewrite the given expression: \[ \left(1+\frac{r}{10^{90}}\right)^{10^{90}}\approx5 \]
02

Take the natural logarithm of both sides

In order to simplify the expression, we will take the natural logarithm of both sides. Using logarithm properties, we can rewrite the equation as follows: \[ 10^{90}\ln\left(1+\frac{r}{10^{90}}\right)\approx\ln 5 \]
03

Simplify the expression and use the limit concept

To simplify the expression, consider the limit as \(x\) approaches infinity of \[ \lim_{x\to\infty}\left(x\ln\left(1+\frac{r}{x}\right)\right) \] Using the limit concept, we know that this expression is equal to \(r\) when \(x=10^{90}\). Therefore, \[ r \approx \ln 5 \]
04

Compute r

Now, we can find the value of \(r\) by computing the natural logarithm of 5. \[ r \approx \ln(5) \approx 1.609 \] So, the value of \(r\) that makes the given expression close to 5 is approximately 1.609.

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