Exponents and Logarithms Definition of a Logarithm Rules Functions Graphs Solving Equations ©Carolyn C. Wheater, 2000

Definition of a Logarithm A logarithm, or log, is defined in terms of an exponential. If bx=a, then logba=x If 52=25 then log525=2 log525=2 is read “the log base 5 of 25 is 2.” You might say the log is the exponent we put on 5 to make 25 ©Carolyn C. Wheater, 2000

Rules for Exponents Exponents give us many shortcuts for multiplying and dividing quickly. Each of the key rules for exponents has an important parallel in the world of logarithms. ©Carolyn C. Wheater, 2000

Multiplying with Exponents To multiply powers of the same base, keep the base and add the exponents. Can’t do anything about the y3 because it’s not the same base. Keep x, add exponents 7 + 5 ©Carolyn C. Wheater, 2000

Dividing with Exponents To divide powers of the same base, keep the base and subtract the exponents. Keep 5, subtract 12-4 Keep 7, subtract 10-6 ©Carolyn C. Wheater, 2000

Powers with Exponents To raise a power to a power, keep the base and multiply the exponents. This means t7·t7·t7 = t7+7+7 ©Carolyn C. Wheater, 2000

Rules for Logarithms Just as the rules for exponents let you easily rewrite a product, quotient, or power, the corresponding rules for logs allow you to rewrite the log of a product, the log of a quotient, or the log of a power. ©Carolyn C. Wheater, 2000

Log of a Product Logs are exponents in disguise To multiply powers, add exponents To find the log of a product, add the logs of the factors The log of a product is the sum of the logs of the factors logbxy = logbx + logby log5(25·125) = log525 + log5125 ©Carolyn C. Wheater, 2000

Log of a Product Think about it: 25·125 = 52 ·53 = 52+3=55 log5(25 ·125) = log5(52 ·53)=log5(52)+log5(53) log525 = log5(52)=2 log5125 = log5(53)=3 log5(25 ·125) = log5(52)+log5(53) = 2 + 3 = 5 log5(25 ·125) = log5(55) =5 Laws of Exponents Logs are Exponents! Add the exponents! ©Carolyn C. Wheater, 2000

Log of a Quotient Logs are exponents To divide powers, subtract exponents To find the log of a quotient, subtract the logs The log of a quotient is the difference of the logs of the factors logb = logbx - logby log5(12525) = log5125 - log525 ©Carolyn C. Wheater, 2000

Subtract the exponents! Log of a Quotient Think about it: 125  25 = 53  52 = 53-2=51 log5(125  25) = log5(53  52) = log5(53) - log5(52) log5125 = log5(53)=3 log525 = log5(52)=2 log5(125  125) = log5(53)-log5(52) = 3 - 2 = 1 log5(125  25) = log5(51) =1 Laws of Exponents Logs are Exponents! Subtract the exponents! ©Carolyn C. Wheater, 2000

Log of a Power Logs are exponents To raise a power to a power, multiply exponents To find the log of a power, multiply the exponent by the log of the base The log of a power is the product of the exponent and the log of the base logbxn = nlogbx log 32 = 2log3 ©Carolyn C. Wheater, 2000

Multiply the exponent by the log (an exponent!) Log of a Power Think about it: 252 =( 52)2 = 52 · 2=54 log5(252) = 2log5(52) log525 = log5(52)=2 log5(252) = 2log5(52) = 2 ·2 = 4 log5(252) = log5(625) = log5(54) = 4 Laws of Exponents Logs are Exponents! Multiply the exponent by the log (an exponent!) ©Carolyn C. Wheater, 2000

Rules for Logarithms The same rules can be used to turn an expression into a single log. logbx + logby = logbxy logbx - logby = logb nlogbx = logbxn ©Carolyn C. Wheater, 2000

Rules for Logarithms A sum of two logs becomes the log of a product. log39 + log327 = log3(9·27) A difference of two logs becomes the log of a quotient. log232 - log28 = log2 A multiple of a log becomes the log of a power 2log57 = log572 Bases must be the same ©Carolyn C. Wheater, 2000

Sample Problem Express as a single logarithm: 3log7x + log7(x+1) - 2log7(x+2) 3log7x = log7x3 2log7(x+2) = log7(x+2)2 log7x3 + log7(x+1) - log7(x+2)2 log7x3 + log7(x+1) = log7(x3·(x+1)) log7(x3·(x+1)) - log7(x+2)2 log7(x3·(x+1)) - log7(x+2)2 = ©Carolyn C. Wheater, 2000

Exponential Functions The exponential function has the form f(x)=abx a is the beginning, or initial amount b is the base, the factor that represents the rate of increase x is the exponent, often representing a period of time ©Carolyn C. Wheater, 2000

Logarithmic Functions The logarithmic function has the form f(x)=logbx b is the base x is the number f(x) is the log (or disguised exponent) ©Carolyn C. Wheater, 2000

Graphs of Exponential Functions The graph of f(x)=bx has a characteristic shape. If b>1, the graph rises quickly. If 0 < b < 1, the graph falls quickly. Unless translated the graph has a y-intercept of 1. 24 ©Carolyn C. Wheater, 2000

Graphs of Logarithmic Functions The graph of f(x)=logbx has a characteristic shape. The domain of the function is {x| x>0} Unless translated, the graph has an x-intercept of 1. -1 1 2 3 4 5 6 ©Carolyn C. Wheater, 2000

Translating the Graphs Both exponential and logarithmic functions can be translated. The vertical and horizontal slides will show up in predictable places in the equation, just as for parabolas and other functions. Shifted 1 unit right and 3 down Shifted 6 units left and 4 up ©Carolyn C. Wheater, 2000

Solving Exponential Equations If possible, express both sides as powers of the same base Equate the exponents Solve ©Carolyn C. Wheater, 2000

Solving Exponential Equations If it is not possible to express both sides as powers of the same base take the log of each side using any convenient base use rules for logs to break down the expressions isolate the variable evaluate and check ©Carolyn C. Wheater, 2000

Solving Exponential Equations Solve Take the log of each side Use rules for logs Isolate the variable Evaluate and check Any convenient base can be used, and since you’ll want to use your calculator, that will probably be 10 x  0.675 ©Carolyn C. Wheater, 2000

Solving Logarithmic Equations Use the rules for logs to simplify each side of the equation until it is a single log or a constant. ©Carolyn C. Wheater, 2000

Solving Logarithmic Equations Log = Log Exponentiate (drop logs) Solve the resulting equation Reject solutions that would mean taking the log of a negative number ©Carolyn C. Wheater, 2000

Solving Logarithmic Equations Log = Constant Use the definition of a logarithm to express as an exponential Evaluate and check ©Carolyn C. Wheater, 2000