Exponential & Logarithmic Functions (DP IB Maths: AA HL)

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5 August 2023

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An exponential function is defined by $f (x) = a^{x}, a > 0$
Its domain is the set of all real values
Its range is the set of all positive real values
An important exponential function is $f (x) = e^{x}$
- Where e is the mathematical constant 2.718…
Any exponential function can be written using e
- $a^{x} = e^{x \ln a}$
  - This is given in the formula booklet

The graphs have a y-intercept at (0, 1)
The graph will always pass through the point (1, a)
The graphs do not have any roots
The graphs have a horizontal asymptote at the x-axis: $y = 0$
- For a > 1 this is the limiting value when x tends to negative infinity
- For 0 < a < 1 this is the limiting value when x tends to positive infinity
The graphs do not have any minimum or maximum points

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A logarithmic function is of the form $f (x) = \log_{a} x, x > 0$
Its domain is the set of all positive real values
- You can't take a log of zero or a negative number
Its range is set of all real values
$\log_{a} x$ and $a^{x}$ are inverse functions
An important logarithmic function is $f (x) = \ln x$
- This is the natural logarithmic function $\ln x \equiv \log_{e} x$
- This is the inverse of $e^{x}$
  - $\ln e^{x} = x$ and $e^{\ln x} = x$
Any logarithmic function can be written using ln
- $\log_{a} x = \frac{\ln x}{\ln a}$ using the change of base formula

The function $f$ is defined by $f (x) = \log_{5} x$ for $x > 0$ .

Write down the inverse of

f

. Give your answer in the form

e^{g (x)}

2-4-2-ib-aa-sl-log-function-a-we-solution

Sketch the graphs of

f

and its inverse on the same set of axes.

2-4-2-ib-aa-sl-log-function-b-we-solution

the (exam) results speak for themselves:

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