Describe the logarithmic function known as \ln x in terms of its base.
Write the following equations in logarithmic form:
Rewrite the following logarithmic equations using index notation:
\ln e = 1
\ln \sqrt{e} = \dfrac{1}{2}
\ln \left(\dfrac{1}{\sqrt{e}}\right) = - \dfrac{1}{2}
\ln 5 = x
Simplify the following:
Is the value of \log_{e} 2 greater than or less than 1?
Consider the following logarithmic expressions:
\log_{e} 7, \log_{3} 7, \log_{2} 7Which expression has the largest value? Explain your answer.
Evaluate each of the following expressions:
\ln e^{3.5}
\ln e^{4}
\sqrt{6} \ln \left(e^{\sqrt{6}}\right)
\ln \left(\dfrac{1}{e^{2}}\right)
Rewrite each of the following expressions as a single logarithm:
\ln 3 + \ln 5
\ln 24 - \ln 4
\ln 8 - \ln 32
\ln 11 + \ln 5 + \ln 8
\ln 24 - \left(\ln 2 + \ln 6\right)
3 \ln \left(x^{5}\right) - 4 \ln \left(x^{2}\right)
6 \ln x - \dfrac{1}{6} \ln y
7 x + \ln \left(1 + e^{ - 7 x }\right)
Simplify each of the following expressions:
\ln 7 + \ln 15
\ln e + \ln e
3 \left(\ln 7 + \ln 3\right)
2 \left(\ln 4 - \ln 2\right)
\dfrac{\ln 49}{\ln 7}
5 \ln 16 - 5 \ln 8
4 \ln 3 - 12 \ln 9
\dfrac{\ln \left(\dfrac{1}{x^{2}}\right)}{\ln x}
\dfrac{\ln a^{6}}{\ln a^{3}}
\ln e + \dfrac{\ln \left(e^{42}\right)}{\ln \left(e^{7}\right)}
\dfrac{48 \ln \sqrt{e}}{\ln \left(e^{6}\right)}
\dfrac{\ln \left(a^{4}\right)}{\ln \sqrt[3]{a}}
Rewrite each of the following as a sum or difference of logarithms:
\ln \left( u v\right)
\ln \left( 3 x\right)
\ln \left( 2 p^2 \right)
\ln \left( x^2 y^7 \right)
\ln \left(\dfrac{19}{7}\right)
\ln \left(\sqrt{\dfrac{c^{8}}{d}}\right)
Use the properties of logarithms to express each of the following without any powers or surds:
Use the properties of logarithms to rewrite each of the following in the form \ln b^{a} where \\a > 0:
Michael has written \ln 25 = \ln \left( 25 \times 1\right) = \ln 25 + \ln 1. Is Michael correct? Explain your answer.
Suppose u = \ln a and v = \ln b. Rewrite the following expressions in terms of u and v:
\ln \left(ab\right)
\ln \left( b^{8} \sqrt{a}\right)
\ln \left(\dfrac{a^{5}}{b^{4}}\right)
\ln \left(\sqrt{\dfrac{a^{7}}{b^{3}}}\right)
Consider x=\ln 31. Find the value of x, correct to two decimal places.
Find the value of each of the following correct to four decimal places:
\ln 94
\ln 0.042
\ln \left( 18 \times 35\right)
\ln \left( 10 - \sqrt 144 \right)
Find the exact value of x in each of the following:
3 \ln x = 9
\ln 3 x = 5
2 \ln 3 x = 6
5 e^{x} = 25
4 e^{x + 7} = 20
\ln \left( 2 x - 1\right) = \ln \left(x + 3\right)
\ln x + 2 = \ln \left( 2 x + 1\right)
e^{ - 5 \ln x } = \dfrac{1}{243}
\ln e^{x} - 3 \ln e = \ln e^{2}
e^{x + \ln 8} = 5 e^{x} + 3
\ln e^{\ln \left(x - 1\right)} - \ln \left(x - 7\right) = \ln 4
e^{ 2 x} - 8 e^{x} + 7 = 0
e^{ 2 x} - 7 e^{x} + 12 = 0
3 e^{ 2 x} - 2 e^{x} = 8
\dfrac{1}{3} e^{ 2 x} + 2 e^{x} = 9
\ln^{2} x - 4 \ln x = 5
Rewrite y = 2 \log_{e} x - 3 with x as the subject of the equation.
Find the value of y in each of the following expressions:
\ln \left(\ln e^{ - y }\right) = \ln 3
e^{ - \ln y } = 3
Consider the function f \left( x \right) = \ln x.
Complete the following table of values, correct to two decimal places:
| x | \dfrac{1}{4} | \dfrac{1}{3} | \dfrac{1}{2} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| f \left( x \right) |
Hence, sketch the graph of f \left( x \right) = \ln x.
State the equation of the vertical asymptote of f \left( x \right) = \ln x.
State the equation of the x-intercept of f \left( x \right) = \ln x.
Consider the graph of the functions \\y = \log_{2} x and y = \log_{3} x:
Use the approximation e = 2.718.
For what values of x will the graph of \\ y = \log_{e} x lie above the graph of \\ y = \log_{3} x and below the graph of \\ y = \log_{2} x?
For what values of x will the graph of \\ y = \log_{e} x lie above the graph of \\ y = \log_{2} x and below the graph of \\ y = \log_{3} x?
Describe the transformation of the following:
g \left( x \right) = \ln x into f \left( x \right) = \ln x + k, where k > 0.
g \left( x \right) = \ln x into f \left( x \right) = \ln x + k, where k < 0.
g \left( x \right) = \ln x into f \left( x \right) = \ln \left(x - h\right), for h > 0.
g \left( x \right) = \ln x into f \left( x \right) = \ln \left(x - h\right), for h < 0.
Consider the following translations:
State the direction in which the graph has been translated.
State the equation of the vertical asymptote of the function.
The graph of y = \ln x is translated to create the graph of y = \ln x + 4.
The graph of y = \ln x is translated to create the graph of y = \ln \left(x + 8\right).
Consider the following graphs of f \left( x \right) = \ln x and g \left( x \right) below:
Describe the transformation required to get from f \left( x \right) to g \left( x \right).
Hence, state the equation of g \left( x \right).
The given graph of f \left( x \right) is the result of two transformations of y = \ln x:
Describe the transformations required for y = \ln x to become f \left( x \right).
Hence state the equation for f \left( x \right).
The given graph f \left( x \right) is the result of two transformations of y = \ln x:
Describe the transformations required for y = \ln x to become f \left( x \right).
Hence state the equation for f \left( x \right).
Consider the function f \left( x \right) = e^{x}.
Sketch the graph of the function, the line y = x and the inverse function f^{-1} \left( x \right) on the same set of axes.
State the kind of function the inverse of f \left( x \right) = e^{x} is.
Hence, state the equation of the inverse function.
For each of the following functions:
Rewrite f \left( x \right) as a sum in simplified form.
Describe how the graph of f \left( x \right) can be obtained from the graph of y = \ln x.
f \left( x \right) = \ln \left( e^{2} x\right)
f \left( x \right) = \ln \left(\dfrac{x}{e^{3}}\right)