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Compound Interest ▼Math Symbols ▲Math Symbols

Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi
  • As in part (a), compute the APR of 10% compounded monthly, biweekly (k = 26), and daily. (We have thrown in the biweekly rate because loans can be paid o? biweekly.)

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Compound Interest ▼Math Symbols ▲Math Symbols

Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi
  • Compute the APR of 5% compounded monthly and daily

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Evaluating an Interesting Limit ▼Hint ▲Hint ▼Math Symbols ▲Math Symbols

Functions like <code>sqrt(x)</code>, <code>ln(x)</code>, and <code>exp(x)</code> are available. Use the helper tool as a reference.
Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

Using \lim_{n \rightarrow \infty} \left(1+ \frac{1}{n}\right)^{n}=e, calculate:

  • \lim_{n \rightarrow \infty} \left(1+ \frac{2}{n}\right)^{5n}

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Evaluating an Interesting Limit ▼Hint ▲Hint ▼Math Symbols ▲Math Symbols

Functions like <code>sqrt(x)</code>, <code>ln(x)</code>, and <code>exp(x)</code> are available. Use the helper tool as a reference.
Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

Using \lim_{n \rightarrow \infty} \left(1+ \frac{1}{n}\right)^{n}=e, calculate:

  • \lim_{n \rightarrow \infty} \left(1+ \frac{1}{n}\right)^{3n}

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Evaluating an Interesting Limit ▼Hint ▲Hint ▼Math Symbols ▲Math Symbols

Functions like <code>sqrt(x)</code>, <code>ln(x)</code>, and <code>exp(x)</code> are available. Use the helper tool as a reference.
Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

Using \lim_{n \rightarrow \infty} \left(1+ \frac{1}{n}\right)^{n}=e, calculate:

  • \lim_{n \rightarrow \infty} \left(1+ \frac{1}{2n}\right)^{5n}

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Solving Equations with e and ln x ▼Math Symbols ▲Math Symbols

Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

We know that the natural log function ln(x) is defined so that if ln(a) = b then e^{a}=a. The common log function log(x) has the property that if log(c) = d then 10^{d}=c. It’s possible to define a logarithmic function \log _{b}(x) for any positive base b so that \log _{b}(e)=f implies b^{f} = e. In practice, we rarely see bases other than 2, 10 and e.

  • Solve for y:


    log(y + 1) = x^{2}+ log(y- 1)

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Solving Equations with e and ln x ▼Math Symbols ▲Math Symbols

Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

We know that the natural log function ln(x) is defined so that if ln(a) = b then e^{a}=a. The common log function log(x) has the property that if log(c) = d then 10^{d}=c. It’s possible to define a logarithmic function \log _{b}(x) for any positive base b so that \log _{b}(e)=f implies b^{f} = e. In practice, we rarely see bases other than 2, 10 and e.

  • Solve for y : 


     \ln(y + 1) + \ln(y - 1) = 2x + \ln

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Solving Equations with e and ln x ▼Math Symbols ▲Math Symbols

Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

We know that the natural log function ln(x) is defined so that if ln(a) = b then e^{a}=a. The common log function log(x) has the property that if log(c) = d then 10^{d}=c. It’s possible to define a logarithmic function \log _{b}(x) for any positive base b so that \log _{b}(e)=f implies b^{f} = e. In practice, we rarely see bases other than 2, 10 and e.

  • Solve for y: 


     2 ln y = ln(y + 1) + x


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Solving Equations with e and ln x ▼Hint ▲Hint ▼Math Symbols ▲Math Symbols

Put  <latex>u = e^{x}</latex>, solve first for u
Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

We know that the natural log function ln(x) is defined so that if ln(a) = b then e^{a}=a. The common log function log(x) has the property that if log(c) = d then 10^{d}=c. It’s possible to define a logarithmic function \log _{b}(x) for any positive base b so that \log _{b}(e)=f implies b^{f} = e. In practice, we rarely see bases other than 2, 10 and e.

  • Solve for x:

    \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=y

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Solving Equations with e and ln x ▼Hint ▲Hint ▼Math Symbols ▲Math Symbols

Try replacing <latex>u=e^{x}</latex> and first solving for <latex>u</latex>. Also, consider only positive values for any square root.
Common forms and functions
  • x+y x+y
  • x·yx*y
  • x/yx÷y
  • xyx**y
  • √xsqrt(x)
  • x!factorial(x)
  • ln(x)log(x)
  • exexp(x)
  • log10(x)log(x, 10)
  • log2(x)log(x, 2)
  • sin(x)sin(x)
  • cos(x)cos(x)
  • tan(x)tan(x)
  • cot(x)1/tan(x)
  • atan(x)atan(x)
  • asin(x)asin(x)
  • acos(x)acos(x)
  • sinh(x)sinh(x)
  • cosh(x)cosh(x)
  • tanh(x)tanh(x)
  • eE
  • πpi

We know that the natural log function ln(x) is defined so that if ln(a) = b then e^{a}=a. The common log function log(x) has the property that if log(c) = d then 10^{d}=c. It’s possible to define a logarithmic function \log _{b}(x) for any positive base b so that \log _{b}(e)=f implies b^{f} = e. In practice, we rarely see bases other than 2, 10 and e.

  • Consider


     y = e^{x}+ e^{-x}


    Solve for x.

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Massachusetts Institute Of Technology
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Logarithmic differentiation; hyperbolic functions

Note: More on "exponents continued" in lecture 7

View the complete course at: http://ocw.mit.edu/18-01F06

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