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

This blog is property of Edward Shore. © 2012

Thanks, my friend. It's a nice approach to Logarithmic and Exponential Functions. It's really helpful. But if you illustrate Expressing many Log Quantities as one Log Quantity. Then it will be a better post.

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