Logarithms Drew Olsen, Chris Ferrer, Matt Sroga, Kevin Gilmartin

Section 4.3 Logarithmic Functions

Logarithmic Functions For b to be greater than 0, b not equal to 0 the inverse f(x) = b^x, denoted f ^ -1(x) = log (base) b X, is the logarithmic function with base b

Logarithmic Functions Forms: –Logarithmic Form y = log (base) b X –Exponential Form x = b^y

Logarithmic Functions The log of the base b of x is the exponent to which b must be raised to obtain x. y = log(base)10 ^x equivalent to x= 10^y y = log(base) e ^x equivalent to x = e^y

Logarithmic Functions Properties: –If b, M, and N are positive real numbers, b does not equal 1, and p and x are real numbers, then… log(base)b 1 = 0 log(base)b b = 1 log(base)b b^x = x b^ log bx = x, x > 0 log(base)b = log(base)b M + log(base)b N log(base)b M/N = log(base)b M - log(base)b N log(base)b M^p = p log(base)b M log(base)b M = log(base)b N only if M=N

Logarithmic Functions Common Logarithms: –Y = log x = log(base)10 x Natural Logarithms: –Y = ln x = log(base)e x Log- Exponential Relationships: Log x = y = x = 10^y Ln x = y = x = e^y

Section 4.4 Logarithmic Models

Logarithmic Models Logarithmic scales can be used to: –compare the intensities of sounds – magnitudes of earthquakes –velocity during rocket flight

Logarithmic Models Sound Intensity: –Decibel - a logarithmic unit of measurement that expresses the magnitude of sound. –D = 10 log I / Io “D”: is the decibel level of sound “I”: is the intensity of the sound “Io” = 10 ^ -12 (threshold of hearing) –Unit = watts per sq meter ( W/m^2)

Logarithmic Models Find the number of decibels from a whisper with the sound intensity of 5.20 X 10^-10 W/m^2. –D = 10 log I / Io = 10 log (5.2 x 10^-10) / (10^-12 ) = 10 log 520 = decibels

Logarithmic Models Earthquake Intensity: –Magnitude - a measure of the energy of an earthquake. –Richter scale - assigns a single number to quantify the amount of seismic energy released by an earthquake. –M = 2/3 log E/Eo E = energy released by earthquake (joules) Eo = 10 ^ 4.40 Joules

Logarithmic Models The 1960 San Francisco earthquake released approximately 5.96 X 10^16 joules of energy. what was its magnitude on the Richer Scale? M = 2/3 log E/Eo –M = 2/3 log (5.96 X 10^16) / 10 ^ 4.40 –M = 8.25

Logarithmic Models Rocket Flight: – Theory of Rocket Flight - uses advanced physics to show that the velocity v of a rocket at burnout (depletion of fuel). –V = c ln Wt / Wb C is the exhaust velocity of the rocket engine Wt is the take off weight Wb is the burnout weight V is the velocity

Logarithmic Models A typical single-stage, solid- fuel rocket may have a weight ratio Wt / Wb = 18.7 and an exhaust velocity c = 2.38 km/s. Would this rocket reach launch velocity of 9.0 km/s? –V = c ln Wt / Wb V = 2.38 In 18.7 V = 6.97 km/s

Section 4.5 Logarithmic Equations

Solving a Logarithmic Equation Algebraic Solution log (x + 3) + log x = 1 log [x (x + 3)] = 1 Combine the left side using log M + log N = log MN x (x + 3) + 10^1 Change to equivalent exponential form x^2 + 3x – 10 = 0 Write the equation “ax^2 + bx + c = 0” form and solve (x + 5) (x – 2) = 0 x = -5, 2 In this case -5 does not work REMEMBER TO CHECK

Solving a Logarithmic Equation Graphical Solution Graph “y = log (x + 3) + log x” and “y = 1” Look at the chart and it will show that the X Value of -5 has a Y Value ERROR while the X Value of 2 has a Y Value of 1. So therefore 2 works while -5 does not.

Real World Solutions Scientist use Logs in a variety of ways, especially to predict the growth of a certain animals over a period of time. An example would be to predict ht growth of a certain type of endangered frog. Consider a pond where there are initially 2 frogs. Every week a pair of frogs pair up and produce 2 more frogs. If the pond has enough food to sustain 1024 frogs, then how many weeks can the frogs keep reproducing before their is not enough food for all of them? 2^n=1024 log[2^n]=log1024 nlog2=log1024 n= {log1024} / {log2} n = 10