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Exponentials and logarithms (The laws of logarithms (Three laws of…
Exponentials and logarithms
The laws of logarithms
x = a^n
n = log a(x)
n is the log of x to base a
When n = 1
a^1 = a
log a(a) = 1
When n = o
a^0 = 1
log a(1) = 0
When n = -1
a^-1 = 1/a
log a(1/a) = -1
Three laws of logarithms
log a(xy) = log a(x)+log a(y)
log a(x/y) = log a(x)-log a(y)
log a(x^k) = k log a(x)
You can write log 10(x) as simply log (x)
Strategy
1 Convert between index notation and logarithmic notation
2 Apply the laws of logarithms if necessary and any results for special cases
3 Manipulate and solve the equation. Check your solution by substituting back into the original equation
Exponential functions
The general equation of an exponential function is y = a^x where a is a positive constant
The graph y = e^x has a gradient of e^x at any point (x, y)
The inverse of y = a^x is the logarithmic function, y = log a(x)
The inverse of y = e^x is log e(x) which can be written as y = ln (x)
ln (x) is called the natural (or Naperian) logarithm
Strategy
1 Draw or sketch a graph if it is helpful
2 Use what you know about the fradients of y =e^x and y = e^kx
3 Use the realtionship between an exponential function and its inverse
Exponential processes
An equation of the form y = Ae^kt gives an exponential model where A and k are constants
Strategy
1 Calculate data using the model
2 Consider sketching or using a graphical calculator to graph the model
3 Use your knowledge of exponential functions and logarithms to find rates of the change and solve equations
4 Compare actual data with your model an, where necessary, comment on any limitations
Curve fitting
y = ax^n becomes Y = nX + c, where Y = log (y), X = log (x)
y = kb^x becomes Y = mx+c, where Y = log (y), m = log (b) and c =log (k)
Strategy
1 Transform the non-linear functions y = ax^n and y =kb^x to linear functions using logarithms
2 Use the transformed data to draw a straight-line graph, using a line of best fit when necessary
3 Use your graph to calculate the constant the constants and work out the realtionship between x and y