Créer une présentation
Télécharger la présentation

Télécharger la présentation
## Logarithmic Functions

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Logarithmic Functions**• Recall that for a > 0, the exponential function f(x) = axis one-to-one. This means that the inverse function exists, and we call it the logarithmic function with base a, written loga. • We have: • Example. log10 1000 = 3, log10 = 0.5, log10 0.01 = –2. • The notation log x is shorthand for log10 x, and log x is called the common logarithm. • The notation ln x is shorthand for loge x, and ln x is called the natural logarithm.**Graphs of 10x 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 logax is one-to-one, it follows that • Since the graphs of loga and logb 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 101/3 = ???**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 log2 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 32x–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 = ax • Domain of the logarithm • Fundamental properties of logarithms log of product log of quotient log of xn • Change of base formula • Solving exponential equations • Solving logarithmic equations