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Logarithmic Functions

Logarithmic Functions

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Logarithmic Functions

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  1. 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.

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

  3. 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,

  4. 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 = ???

  5. Write the expression as a single logarithm

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

  7. 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:

  8. Exponential Equations • When solving an exponential equation, consider taking logarithms of both sides of the equation. • Example. Solve 32x–1 = 17.

  9. 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.

  10. 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.

  11. 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