Math 1431 Section 4.3 Section 4.3: Logarithmic Function

The exponential function is 1-1; therefore, it has an inverse function. The inverse function of the exponential

function with base a is called the logarithmic function with base a. The function fx ( ) loga x is the logarithmic function with base a with x > 0, a > 0 and a  1. For example, for a > 1:

 If a = 10 then fx() log x. This is known as the common base.

 If a = e then fx() ln x. This is known as the natural logarithm.

 ln(0) is undefined.

 ln(1) 0

Graphs The orientation of the logarithmic function depends on its base, 0 < a < 1 or a > 1.

Both graphs have a vertical asymptote of x = 0 (the y-axis). Their domain is 0, and range is ,  .

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Math 1431 Section 4.3 Example 1: Find the domain of yx ln(ln( )) .

The logarithmic function with base a satisfies: x  loga ax for all x ,  and log x  axa  , for all x 0,

Examples:

3 log2=2

ln e =

eln 5 =

3log3 x =

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Math 1431 Section 4.3 Converting Expressions

y Fact: y  loga x is equivalent to ax

For example: 5 2 32 is equivalent to log2  32 5

2 log4  16 2 is equivalent to 16 4

Laws of Logarithms

Product Rule for Logs: logaaa x  yxy  log log

x Quotient Rule for Logs: logaaa logx log y y p Power Rule for Logs: logaa x  px log

Example 2: Expand using the Laws of Logarithms:

x3y ln  zx1

1 2 Example 3: Simplify using the Laws of Logarithms: logx  logxz 1 4log 3  

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Math 1431 Section 4.3 Differentiating Natural Logarithmic Function: d 1 ln x  dx x If u is a function of x , then use the chain rule.

Example 4: Differentiate the following functions. a. yxxln 43 5 

b. yx ln sin

c. f xx ln ln  2 

Sometimes it’s easier to use the rules of logarithms to differentiate a log function. 4

Math 1431 Section 4.3

x4 Example 5: Differentiate . Then find the equation of the tangent line at the point gx() ln2 31x  1 1, ln . 2

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Math 1431 Section 4.3 Example 6: Determine where f ()xxx ln is increasing/decreasing, any relative extrema, concavity, and any points of inflection.

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Math 1431 Section 4.3 Different Bases ln x Change of base formula: log x  a ln a d 11 log x  dxa xln a If u is a function of x , then use the chain rule.

Example 7: Find the derivative of each function. a. yx log2

3 b. yxxlog3  4 2

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Math 1431 Section 4.3 log x c. fx() 9 x2

We know how to find the derivative of functions such as:

 x5 ---use the power rule

 5x ---use exponential rule

But how about xx ? We will use a method called Logarithmic Differentiation.

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Math 1431 Section 4.3

Example 8: Find the derivative of yx tan x .

Step 1: Take natural logs of both sides of the equation.

Step 2: Use any rules of logs to simplify the equation.

Step 3: Take the derivative of both sides of the equation.

Step 4: Solve for y ' . Remember what y was equal to originally.

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Math 1431 Section 4.3

Example 9: Find the derivative of yx 21x . Step 1: Take natural logs of both sides of the equation.

Step 2: Use any rules of logs to simplify the equation.

Step 3: Take the derivative of both sides of the equation.

Step 4: Solve for y ' . Remember what y was equal to originally.

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Math 1431 Section 4.3 Try these: Find the derivative of: yx log cos 4 .

Let f  xx  ln cos 2 x , find f '  .

2 x Determine where fx() 2 x ln is increasing/decreasing. 4

2 x Find the points of inflection for the function fx() 4 x ln. 4

2x Find the derivative of f xe ln(2 x ) .

x2 x Find the derivative of fx ln 5 . 

4sin x Find the slope of the tangent line to the curve yx2cos at x  2 .

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