Logarithmic function
You already know how to calculate a logarithm. Now it is time to study it as a function: what does its graph look like? Where is it defined? How is it related to the exponential function you already know? The short answer is that the logarithm undoes the power, which means that its graph is that of the exponential function... seen in a mirror.
Domain and vertical asymptote
Determine the domain of definition of the following logarithmic functions: a) $f(x) = \log(x - 5)$ b) $g(x) = \ln(2x + 8)$
Determine the domain of definition of the following logarithmic functions:
a) $f(x) = \log(x - 5)$
b) $g(x) = \ln(2x + 8)$
Increasing or decreasing depending on the base
State, giving reasons for your answer and without sketching them, which of the following functions are increasing and which are decreasing: a) $f(x) = \log_3 x$ b)…
State, giving reasons for your answer and without sketching them, which of the following functions are increasing and which are decreasing:
a) $f(x) = \log_3 x$
b) $g(x) = \log_{0,5} x$
c) $h(x) = \ln x$
Look at the graphs of the logarithmic functions shown in the image above. Match, giving reasons for your answer, Function A and Function B with their respective…
Look at the graphs of the logarithmic functions shown in the image above. Match, giving reasons for your answer, Function A and Function B with their respective algebraic expressions:
1) $y = \log_2 x$
2) $y = \log_{1/2} x$
Inverse of the exponential function
The image shows the graph of the exponential function $f(x) = e^x$ and its inverse function $g(x)$. a) Write the algebraic expression for $g(x)$. b) Explain the…
The image shows the graph of the exponential function $f(x) = e^x$ and its inverse function $g(x)$.
a) Write the algebraic expression for $g(x)$.
b) Explain the geometric meaning of the dashed orange line $y = x$ in relation to both curves.
c) Determine the domain and range of both functions, justifying how they swap between a function and its inverse.
Translations: $y=\log_a(x+b)$
Given the function $f(x) = \log_3(12 - 3x)$: a) Determine its domain algebraically. b) State the equation of its vertical asymptote. c) In which direction (left or…
Given the function $f(x) = \log_3(12 - 3x)$:
a) Determine its domain algebraically.
b) State the equation of its vertical asymptote.
c) In which direction (left or right) does the graph of this function open relative to its asymptote?
From the graph of the function $f(x)$ shown above, which corresponds to a translated logarithm of the form $f(x) = \log_a (x + b)$: a) State the value of $b$ by…
From the graph of the function $f(x)$ shown above, which corresponds to a translated logarithm of the form $f(x) = \log_a (x + b)$:
a) State the value of $b$ by observing the vertical asymptote.
b) Use the point $B(1, 2)$ to calculate the base $a$ of the logarithm.
c) Write the full algebraic expression for the function.
Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x - 5)$. Express the result in interval notation.
Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x - 5)$. Express the result in interval notation.
It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin $(0, 0)$ and through the point $(6, 1)$. Set up a system of…
It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin $(0, 0)$ and through the point $(6, 1)$.
Set up a system of equations to find the values of the parameters $a$ and $b$. Once found, write the final algebraic expression for the function and state its vertical asymptote.