## Wednesday, April 25, 2012

### Calculus Revisited #3: Logarithmic and Exponential Functions

Welcome to Part 3 of the 21 part series: Calculus Revisited. Today will we talk about exponential and logarithmic functions.

Exponential and Logarithmic Functions

The function y(x) = e^x is a function where the constant e is taken to the power of x. To 25 decimal places, e = 2.7182818284590452353602874. e is known as the Euler's Number or Napier's Constant. (Wikipedia Source)

The function y(x) = ln x is known as the natural logarithmic function (base e), and is the inverse function of the exponential function. In computer mathematical software and Microsoft Excel ln x is referred to as log x.

Note: On calculators, the function log x refers to a common logarithmic function, not natural. log x uses base 10.

Graphs of both e^x and ln x follow.

Common Properties of the Exponential and Logarithmic Functions

e^x * e^y = e^(x + y)
e^x / e^y = e^(x- y)
(e^x)^y = e^(x * y)

ln(x * y) = ln x + ln y
ln(x / y) = ln x - ln y
ln(x^y) = y * ln x

e^(ln x) = x
ln (e^x) = x

log_n θ = ln θ / ln n (logarithm to base n)

Problems
1. Solve 2^x = 68

2^x = 68
Take the logarithm of both sides
ln (2^x) = ln 68
x ln 2 = ln 68
x = ln 68/ln 2 ≈ 6.08746

2. Solve e^(2x) = 2 * e^(3x)

e^(2x) = 2 * e^(3x)
e^x ≠ 0 for all x, so we can divide.
Divide by e^(2x)
1 = 2 * e^(3x) / e^(2x)
1 = 2 * e^(3x - 2x)
1 = 2 * e^x
1/2 = e*^x
x = ln (1/2) ≈ -.69315

3. Solve e^(2x + 1) = 2^(2x - 3)

e^(2x + 1) = 2^(2x - 3)
ln e^(2x + 1) = ln 2^(2x - 3)
ln e = 1
2x + 1 = (2x - 3) * ln 2
2x + 1 = (2 ln 2)x - (3 ln 2)
(2 - 2 ln 2)x = -3 ln 2 - 1
(2 - 2 ln 2)x = -(3 ln 2 + 1)
x = -(3 ln 2 + 1)/(2 - 2 ln 2) ≈ -5.01778

Next time, we tackle limits. Until next time, take care! Eddie

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