2.9 Logarithmic expressions
Consider logarithmic and exponential expressions:
Write the logarithmic expression \log_{b} c = a in exponential form and describe the relationship between b, \, c, and a in this expression.
Conversely, if you have the exponential expression b^{a} = c, how would you convert it into its equivalent logarithmic form?
What is the base of a natural logarithm and how is it typically represented?
What is the base of a logarithm if it is not specified in a logarithmic expression?
For each equation, identify the:
base
exponent
argument
What is the difference between a standard scale and a logarithmic scale?
Write down the first 5 units of a logarithmic scale with a base of 10.
Calculate the value of the logarithmic expressions:
\log_{2} 16
\log_{3} 81
\log_{5} 125
\log_{10} 10000
Rewrite each logarithmic expression in exponential form:
\log_{b} a = 3
\log_{m} 5 = n
\log_{4} 64
\log_{3} 27
Convert the exponential equations into their equivalent logarithmic forms:
b^4 = a
3^{4} = 81
5^{3} = 125
7^{2} = 49
For each of the following equations:
Rewrite the equation in logarithmic form.
Approximate the value of x to two decimal places.
Solve for x:
\log_{2} x = 8
\log_{x} 64 = 2
\log_{10} x = 2
\log_{3} (x-1) = 4
Express the numerical values in terms of the indicated exponential expressions:
1000 in terms of 10^x
27 in terms of 3^x
\dfrac{1}{8} in terms of 2^x
\dfrac{1}{125} in terms of 5^x
Use the properties of logarithms to simplify the expressions:
If the pH of a solution is measured on a logarithmic scale and the pH is 6, what is the actual hydrogen ion concentration of the solution?
If a graph uses a logarithmic scale and the units are 1, \, 2, \, 3, what does this mean in terms of the base of the logarithm?
What would be the value of the fifth unit on the logarithmic scale if the base of a logarithm is 10?
How would the units on a logarithmic scale with a base of 2 compare to the units on a standard scale?
Solve for x: \, 5^{2x+1} = 125, express your answer in logarithmic form.
Estimate the value of \log_{3} 500 using a calculator and explain the steps you took to arrive at the approximation.
The Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale with base 10. An earthquake that measures 5 on the Richter scale is 10 times more powerful than an earthquake that measures 4. If an earthquake measures 7 on the Richter scale, how much more powerful is it than an earthquake that measures 5?
A certain city's population P (in thousands) is modeled by the equation P = 500 \log_{10} (t + 1), where t is the time in years since the city was founded. If the population was 1500 thousand, how many years had passed since the city was founded?
The loudness of a sound in decibels is given by the formula L = 10 \log_{10} \dfrac{I}{I_0}, where I is the intensity of the sound and I_0 is a constant reference intensity. If a sound is 10 decibels louder than another sound, how many times more intense is it?
In the context of pH levels in chemistry, which is a logarithmic scale, a solution with a pH of 3 is ten times more acidic than a solution with a pH of 4. If a solution has a pH of 2, how many times more acidic is it than a solution with a pH of 5?