Logarithms
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Logarithms the inverse of exponential functions The logarithmic functions help us work easily with very large or very small numbers…. While calculators have helped us do this, notice that the LOG and In buttons are STILL a part of the calculator and are still an important part of higher mathematics. Remember how we had to determine the x intercepts for some exponential graphs with trial and error? Logs will deliver us from this! I like to think of logs as exponents because of the following…. We must become masters of translating an exponential expression into logarithmic and visa versa. πͺπππππ ππ: ππ = ππ This is what we call “ exponential form”. Let’s change it to “logarithmic form”. log4 64 = 3 Look closely how that translation went… π π = ππ log4 64 = 3 The exponent becomes what the log expression is π»πππ ππ ππππ : ππππ πtoππ ππ ππ π equal to! See whyπππ I saidπππ logs are equal exponents. The BASE in the exponential expression becomes the BASE in the logarithmic expression. You try it! π π = ππ log2 32 = 5 You need to translate in the opposite direction too! π log2 π π −π = -3 π = π Your calculator will ONLY calculate logs base 10. Log is called the “common log”. It is so common that when we are referring to log base 10 we don’t include the base. log2 x ππππ ππ ππππ π log 14 log5 14 ππππ ππ ππππ π πππ πππ πππ′ π ππππππππ ππππ ππ ππππ ππππππππππ. πππ πππ ππππππππ ππππ ππ ππππ ππππππππππ. . log x ππππ ππ ππππ ππ Your calculator will also calculate logs base e but it uses a different button. ( In) This log is called the natural log and must be used with e. ln x ππππ ππ ππππ π ln 45 Practice! EVALUATE: log2 8 It helps to write it into exponential form. log2 8 = x π π =8 So 2 to WHAT POWER results in 8? Practice! EVALUATE: log36 6 It helps to write it into exponential form. log36 6 = x π ππ = 6 So 36 to WHAT POWER results in 6? Practice! EVALUATE: log5 0.2 It helps to write it into exponential form. log5 0.2 = x π π = 0.2 So 5 to WHAT POWER results in 0.2? Practice! EVALUATE: ππππ 125 π ππππ 125 = x π π π = π 125 Practice! EVALUATE: log 100 It helps to write it into exponential form. log 100 = x π ππ = 100 So 10 to WHAT POWER results in 100? Practice! EVALUATE: π ππ ππ π©ππππ πππ πππ ππππ it simplifies to 20! Find the inverse of the function. π = ππππ π Rewrite it in exponential form…. ππ π = π = ππ THEN switch the x and y…. Find the inverse of the function. π ππ= +πππ π= π π (π-3) Rewrite it in exponential form…. π π =π−π THEN switch the x and y…. π π =π−π THEN solve for y Find the inverse of the function. π π−π = π πππ+π (π-3) π = π+ 2 Move the 2 over…. π − π = ππππ (π-3) Rewrite it into exponential form…. π−π π =π−π Switch x and y and solve for y.
