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Logarithmic functions

Logarithmic functions. STUDENTS WILL BE ABLE TO: CONVERT BETWEEN EXPONENT AND LOG FORMS SOLVE LOG EQUATIONS OF FORM LOG B Y=X FOR B, Y, AND X. Warm up: Solve the following. 1. 4 6 =4 3x 2. 8=x 3 3. 27=3 x 4. x 1/4 =2. Inverses of exponential functions.

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Logarithmic functions

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  1. Logarithmic functions STUDENTS WILL BE ABLE TO: CONVERT BETWEEN EXPONENT AND LOG FORMS SOLVE LOG EQUATIONS OF FORM LOGBY=X FOR B, Y, AND X

  2. Warm up: Solve the following • 1. 46=43x • 2. 8=x3 • 3. 27=3x • 4. x1/4=2

  3. Inverses of exponential functions • We know what exponential functions look like as equations and as graphs • Let’s look at f(x)=2x • The inverse of f(x)=2x is a base 2 logarithmic function. • f-1(x)=log2x  (“log base 2 of x”) • They share all the characteristics of inverses. • Reflection over the line y=x • Points are flipped

  4. Logarithms • A logarithm to the base b of a positive number yis defined as follows: • If , then . • We know that a positive number b raised to any power x cannot equal a number y that is less than or equal to zero. Therefore, the log of a negative number or zero is undefined. Read as “log base b of y”

  5. Converting • To convert between logarithmic and exponential form, just substitute the values into the appropriate places. • If , then . • If , then .

  6. Evaluating Logarithms • Evaluate • Step 1: Write as an equation • Step 2: Convert to exponential form • Step 3: Write each side using the same base • Step 4: Power Property of Exponents • Step 5: Set the exponents equal • Step 6: Solve for x.

  7. Common logarithms • A log in base 10 is called a common logarithm. • We can write as • The “log” button on your calculator evaluates common logs only

  8. Solving log equations • This looks weird to most people (myself included!) so to solve this, we convert to exponential form. • Now it’s easy to solve! • Convert to exponential form. • Not quite as easy to solve, but still doable! • Base must be positive! • Convert to exponential form • Solve for x.

  9. Your turn!

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