Logarithmic Functions Recall that for a > 0, the exponential function f(x) = a x is one-to-one. This means that the inverse function exists, and we call it the logarithmic function with base a, written log a. We have: Example. log = 3, log 10 = 0.5, log = –2. The notation log x is shorthand for log 10 x, and log x is called the common logarithm. The notation ln x is shorthand for log e x, and ln x is called the natural logarithm.

Graphs of 10 x and log x  Note that the domain of log x is the set of positive x values.

Logarithmic Identities and Properties From the definition of the logarithm it follows that From the fact that the exponential and logarithmic functions are inverse functions it follows that Since log a x is one-to-one, it follows that Since the graphs of log a and log b intersect only at x = 1,

Fundamental Properties of Logarithms The following three properties of logarithms can be proved by using equivalent exponential forms. Problems. log 2 + log 5 = ???, log 250 – log 25 = ???, log 10 1/3 = ???

Write the expression as a single logarithm

Solving an equation using logarithms If interest is compounded continuously, at what annual rate will a principal of $100 triple in 20 years?

Change of Base Formula Sometimes it is necessary to convert a logarithm to base a to a logarithm to base b. The following formula is used: Compute log 2 27 using common logs and your calculator. Check your answer:

Exponential Equations When solving an exponential equation, consider taking logarithms of both sides of the equation. Example. Solve 3 2x–1 = 17.

Solving an Exponential Equation for Continuous Compounding Problem. A trust fund invests $8000 at an annual rate of 8% compounded continuously. How long does it take for the initial investment to grow to $12,000? Solution. We must solve for t in the following equation.

Logarithmic Equations When solving a logarithmic equation, consider forming a single logarithm on one side of the equation, and then converting this equation to the equivalent exponential form. Be sure to check any "solutions" in the original equation since some of them may be extraneous. Problem. Solve for x.

Summary of Exponential and Logarithmic Functions; We discussed Definition of logarithm as inverse of exponential The common logarithm The natural logarithm Graphs of y = log x and y = a x Domain of the logarithm Fundamental properties of logarithms  log of product  log of quotient  log of x n Change of base formula Solving exponential equations Solving logarithmic equations