Exponents and Logarithms
Maryann Crisci
Mrs. Cappiello
Algebra 2/Trig, Period 6
1 April 2012
Exponents and Logarithms An exponent is the number representing the power a given number is raised to. Exponential functions are used to either express growth or decay. When a function is raised to a positive exponent, it will cause growth. However, when a function is raised to a negative exponent, it will cause decay. Logarithms work differently than exponents. Logarithms represent what power a base should be raised to in order to produce a specific given number. Logarithmic and exponential functions are often used together because they are inverse functions, and therefore “undo” one another. The natural exponential function and the natural logarithm are frequent examples. These functions have a base of the constant e, or Euler’s number. The natural exponential function is written as f(x)=ex. Its inverse, the natural logarithmic function, is written as f(x)=ln(x). The decay of radioactive substances can be represented by using an exponential function. The mass of the radioactive substance will decrease as time passes, but the rate of decay and the mass of the substance will always remain directly proportional. The decay of radioactive substances can be expressed by the function m(t)=m0e-rt, where m(t) represents the mass remaining at time t, r represents the rate of decay, and m0 represents the initial mass. When the rate of decay is expressed by half-life, the rate of decay expressed as a proportion of the mass can be found by substituting the half-life into the equation ½=1∙e-rh. This is because when h, or the half-life, is equal to t, or time, then the mass of 1 unit becomes ½ unit. An example of a radioactive substance that will decay over time is Polonium-210. Polonium-210 has a half life of 140 days. This half-life was plugged into the equation ½=1∙e-rh in order to find the rate of decay expressed as a proportion of the mass, or r. 140 was plugged in for h. In order to solve the
Mrs. Cappiello
Algebra 2/Trig, Period 6
1 April 2012
Exponents and Logarithms An exponent is the number representing the power a given number is raised to. Exponential functions are used to either express growth or decay. When a function is raised to a positive exponent, it will cause growth. However, when a function is raised to a negative exponent, it will cause decay. Logarithms work differently than exponents. Logarithms represent what power a base should be raised to in order to produce a specific given number. Logarithmic and exponential functions are often used together because they are inverse functions, and therefore “undo” one another. The natural exponential function and the natural logarithm are frequent examples. These functions have a base of the constant e, or Euler’s number. The natural exponential function is written as f(x)=ex. Its inverse, the natural logarithmic function, is written as f(x)=ln(x). The decay of radioactive substances can be represented by using an exponential function. The mass of the radioactive substance will decrease as time passes, but the rate of decay and the mass of the substance will always remain directly proportional. The decay of radioactive substances can be expressed by the function m(t)=m0e-rt, where m(t) represents the mass remaining at time t, r represents the rate of decay, and m0 represents the initial mass. When the rate of decay is expressed by half-life, the rate of decay expressed as a proportion of the mass can be found by substituting the half-life into the equation ½=1∙e-rh. This is because when h, or the half-life, is equal to t, or time, then the mass of 1 unit becomes ½ unit. An example of a radioactive substance that will decay over time is Polonium-210. Polonium-210 has a half life of 140 days. This half-life was plugged into the equation ½=1∙e-rh in order to find the rate of decay expressed as a proportion of the mass, or r. 140 was plugged in for h. In order to solve the
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