Determine the inverse of the function. Express your answers in log form.
\displaystyle
y = 4^x
Determine the inverse of the function. Express your answers in log form.
\displaystyle
y = a^x
Determine the inverse of the function. Express your answers in log form.
\displaystyle
y = (\frac{3}{4})^x
Determine the inverse of the function. Express your answers in log form.
\displaystyle
m = p^q
Describe the transformations that must be applied to the parent function y= \log x to obtain each of the following functions.
f(x) = - \log(2x)
Describe how the graphs of f(x) = \log x and g(x) = 3\log(x - 1) + 2 are similar yet different.
Evaluate.
\displaystyle
\log_7343
Evaluate.
\displaystyle
\log_{\frac{1}{5}}25
Evaluate.
\displaystyle
\log_{19}1
Evaluate.
\displaystyle
\log_{4}\frac{1}{256}
Estimate the value to three decimal places.
\displaystyle
\begin{array}{llllllll}
&(a) \phantom{.} \log_353
&(b) \phantom{.} \log_4 \frac{1}{10} \\
&(c) \phantom{.} \log_6 159
&(d) \phantom{.} \log_{15} 1456
\end{array}
Express as a single logarithm.
\displaystyle
\log 5 + \log 11
Express as a single logarithm.
\displaystyle
\log 20 - \log 4
Express as a single logarithm.
\displaystyle
\log_56 + \log_58-\log_512
Express as a single logarithm.
\displaystyle
2\log 3 + 4\log 2
Evaluate using log laws.
\displaystyle
\log_642 - \log_67
Evaluate using log laws.
\displaystyle
\log_35 + \log_318 -\log_310
Evaluate using log laws.
\displaystyle
\log_7 \sqrt[3]{49}
Evaluate using log laws.
\displaystyle
2 \log_48
Describe how the graph of y = \log(10 000x) is related to the graph of y = \log x.
Solve for x.
\displaystyle
5^x = 3125
Solve for x.
\displaystyle
4^x = 16 \sqrt{128}
Solve for x.
\displaystyle
4^{5x} =16^{2x-1}
Solve for x.
\displaystyle
3^{5x} 9^{x^2} = 27
Solve. Express each answer to three decimal places.
\displaystyle
6^x = 78
Solve. Express each answer to three decimal places.
\displaystyle
5.8^x = 234
Solve. Express each answer to three decimal places.
\displaystyle
8\cdot 3^x = 132
Solve. Express each answer to three decimal places.
\displaystyle
200 \cdot (1.23^x) = 540
Solve.
\displaystyle
4^x + 6 \cdot 4^{-x} = 5
Solve.
\displaystyle
8 \cdot 5^{2x} + 8 \cdot 5^x =6
The half-life of a certain substance is 3.6 days. How long will it take for 20 g of the substance to decay to 7 g?
Solve.
\displaystyle
\log_5(2x - 1) =3
Solve.
\displaystyle
\log 3x = 4
Solve.
\displaystyle
\log_4(3x - 5) = \log_411 + \log_42
Solve.
\displaystyle
\log(4x - 1) = \log(x + 1) + \log 2
Solve.
\displaystyle
\log(x + 9) - \log x = 1
Solve.
\displaystyle
\log x + \log(x -3) = 1
Solve.
\displaystyle
\log(x - 1) + \log(x + 2) = 1
Solve.
\displaystyle
\log\sqrt{x^2 -1} = 2
Recall that L = 10 \log (frac{I}{I_0}), where I is the intensity of sound in watts per square metre (W/m^2) and I_0 = 10^{12}W/m^2. Determine the intensity of a baby screaming if the noise level is 100 dB.
What is the sound intensity in watts per square metre (W/m^2) of an engine that is rated at 82 dB?
How many times more intense is an earthquake of magnitude 6.2 than an earthquake of magnitude 5.5?
Pure water has a pH value of 7.0. How many times more acidic is milk, with a pH value of 6.4, than pure water?
Does an increase in acidity from pH 4.7 to pH 2.3 result in the same change in hydrogen ion concentration as a decrease in alkalinity from 12.5 to 10.1? Explain.
Is an exponential model appropriate for the data in the following table? If it is, determine the equation that models the data.
The population of a town is decreasing at the rate of 1.6%/a. If the population today is 20 000, how long will it take for the population to decline to 15 000?
The population of a town is decreasing at the rate4 of 1.6%/a. If the population today is 20 000, how long will it take for the population to decline to 15 000?
a) Calculate the average rate of growth over the entire time period.
b) Calculate the average rate of growth for the first 30 years. How does it compare with the rate of growth for the entire time period?
c) Determine an exponential model for the data.
d) Estimate the instantaneous rate of growth in
- i) 1970
- ii) 1990
The following data show the number of people (in thousands) who own a DVD player in a large city or linear is best for over a period of years.
a) Determine if an exponential or linear model is best for this data.
b) Use your model to predict how many people will own a DVD player in the year 2015.
c) What assumptions did you make to make your prediction in part b)? Do you think this is reasonable? Explain.
d) Determine the average rate of change in the number of DVD players in this city between 1999 and 2002.
e) Estimate the instantaneous rate of change in the number of DVD players in this city in 2000.
f) Explain why using an exponential model to answer part b) does not make sense.